#7654757

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	#34963	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4906 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:

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

Polycyclic presentation:

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

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, 7561953);