#7658453

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	#42805	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4563 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,e^2=ghi,f^2=hi,b^a=bg,c^a=ch,d^a=di,c^b=ci,d^b=dhi,f^e=fhi

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, 7565649);