#8104705

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5813 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)(3,5)(4,7)(6,8)(9,10)(11,16)(12,15)(13,14)(17,18)(19,27)(20,24)(21,26)(22,25)(23,29)(28,30)(31,32);(1,3,5,2)(4,7,8,6)(10,14)(11,12)(17,19,22,29)(18,23,25,27)(20,28,26,32)(21,30,24,31);(1,4,5,8)(2,6,3,7)(9,11,13,12)(10,15,14,16)(17,20,22,26)(18,24,25,21)(19,28,29,32)(23,31,27,30);(2,3)(4,8)(9,12,13,11)(10,16,14,15)(18,25)(20,26)(23,27)(28,32);(2,3)(6,7)(10,14)(15,16)(19,29)(23,27)(28,32)(30,31);(17,21,22,24)(18,26,25,20)(19,30,29,31)(23,32,27,28)

Polycyclic presentation:

2^9|b^2=gi,c^2=g,d^2=i,f^2=gh,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72282980, #72304020, #72304029
maximal normal subgroup of	#72282980, #72304020, #72304029
maximal quotient of	#72304076, #72304081

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8011901);