#8757712

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5201 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=g,d^2=g,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^c=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72311493, #72318949, #72319710, …
maximal normal subgroup of	#72311493, #72318949, #72319710, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8664908);