#8095028

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

4051 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8002224);