#9040991

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4388 conjugacy classes of subgroups; only the 2854 normal subgroups are shown.

normal subgroups only

This group has 4388 conjugacy classes of subgroups — too many to draw. Only its 2854 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,11)(3,8,12,6)(4,9,13,16)(5,10,14,15)(17,18)(19,27)(20,30)(21,26)(22,32)(23,25)(24,29)(28,31)(33,34,37,41)(35,38,43,39)(36,40,45,48)(42,46,44,47);(1,3,7,12)(2,8,11,6)(4,14,13,5)(9,15,16,10)(17,19,23,29)(18,24,25,27)(20,31,32,26)(21,22,28,30)(33,35)(34,38)(36,44)(37,43)(39,41)(40,47)(42,45)(46,48);(3,12)(5,14)(6,8)(10,15)(18,25)(21,28)(22,30)(24,27)(33,35,37,43)(34,39,41,38)(36,42,45,44)(40,47,48,46);(1,4)(2,9)(3,5)(6,15)(7,13)(8,10)(11,16)(12,14)(17,20)(18,22)(19,26)(21,24)(23,32)(25,30)(27,28)(29,31)(33,36)(34,40)(35,42)(37,45)(38,46)(39,47)(41,48)(43,44);(1,5)(2,10)(3,13)(4,12)(6,9)(7,14)(8,16)(11,15)(17,21,23,28)(18,26,25,31)(19,22,29,30)(20,27,32,24)(33,35)(34,38)(36,44)(37,43)(39,41)(40,47)(42,45)(46,48);(1,6)(2,3)(4,15)(5,9)(7,8)(10,13)(11,12)(14,16)(17,22)(18,20)(19,28)(21,29)(23,30)(24,31)(25,32)(26,27)(33,34)(35,38)(36,40)(37,41)(39,43)(42,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=gh,e^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eghi,e^c=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8948187);