#8096961

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

5966 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210246
maximal normal subgroup of	#72210246
maximal quotient of	#72210242, #72266996, #72266999, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004157);