#8932231

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

5260 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)(10,13)(11,15)(17,18,22,23)(19,24,27,31)(20,25,29,28)(21,26,32,30);(3,7)(6,8)(9,10,14,13)(11,16,15,12)(18,23)(20,29)(24,31)(26,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,12)(13,16)(14,15)(19,27)(21,32)(24,31)(26,30);(9,10,14,13)(11,12,15,16)(17,19,22,27)(18,24,23,31)(20,30,29,26)(21,28,32,25);(1,3)(2,6)(4,7)(5,8)(9,12,14,16)(10,15,13,11)(17,20)(18,25)(19,26)(21,31)(22,29)(23,28)(24,32)(27,30);(1,2)(3,6)(4,5)(7,8)(9,13,14,10)(11,16,15,12)(17,21,22,32)(18,26,23,30)(19,28,27,25)(20,31,29,24)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=g,e^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dgi,f^c=fgi,e^d=egi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318286, #72318304, #72319378
maximal normal subgroup of	#72318286, #72318304, #72319378

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8839427);