#7674184

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	#34636	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7581380);