#8093263

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

6434 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,e^2=ghi,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72258185, #72258259, #72258262, …
maximal normal subgroup of	#72258185, #72258259, #72258262, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000459);