#7678266

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	#43011	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4748 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2,6,4)(3,7,5,8)(9,10,13,16)(11,14,12,15)(17,18)(19,23)(20,28)(21,25)(22,27)(24,29)(26,30)(31,32);(1,3,6,5)(2,7,4,8)(9,10)(11,14)(12,15)(13,16)(17,18,21,25)(19,23,27,22)(20,24,29,28)(26,31,32,30);(2,4)(7,8)(17,19,21,27)(18,22,25,23)(20,26,29,32)(24,30,28,31);(1,4,6,2)(3,8,5,7)(9,11,13,12)(10,14,16,15)(17,19)(18,23)(20,26)(21,27)(22,25)(24,31)(28,30)(29,32);(9,11,13,12)(10,14,16,15)(17,18)(19,23)(20,24)(21,25)(22,27)(26,31)(28,29)(30,32);(1,5,6,3)(2,8,4,7)(9,12,13,11)(10,15,16,14)(17,20)(18,24)(19,26)(21,29)(22,30)(23,31)(25,28)(27,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=i,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=ei

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585462);