#8994068

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

5934 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72340421, #72340424
maximal normal subgroup of	#72340421, #72340424

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8901264);