#7654599

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	#2403	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5035 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)(11,15)(12,16)(13,14)(17,18)(19,22)(20,30)(21,25)(23,26)(24,29)(27,32)(28,31);(1,2)(3,7)(4,8)(5,6)(10,14)(11,16)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,32)(30,31);(9,11)(10,12)(13,16)(14,15)(17,20,23,21)(18,25,26,30)(19,27,29,28)(22,31,24,32);(9,12)(10,11)(13,15)(14,16)(17,19)(18,24)(20,27)(21,28)(22,26)(23,29)(25,32)(30,31);(1,3,5,4)(2,7,6,8)(17,21,23,20)(18,25,26,30)(19,28,29,27)(22,31,24,32);(1,4,5,3)(2,8,6,7)(17,22,23,24)(18,19,26,29)(20,31,21,32)(25,27,30,28)

Polycyclic presentation:

2^9|c^2=hi,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, 7561795);