#6995245

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1867 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257364, #72257398, #72264300
maximal normal subgroup of	#72257364, #72257398, #72264300
maximal quotient of	#72308355

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6902441);