#10363692

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4310 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10270888);