#6998927

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
prime radical	#24248	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1658 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)(19,29)(20,26)(21,27)(22,28)(23,25)(24,30)(31,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,24)(20,23)(21,31)(22,30)(25,32)(26,27)(28,29);(9,11)(10,13)(12,16)(14,15)(18,25)(20,31)(21,23)(24,32)(26,29)(27,28);(2,5)(6,8)(11,16)(13,15)(17,20)(18,26)(19,21)(22,31)(23,30)(24,25)(27,29)(28,32);(1,3,4,7)(2,6,5,8)(10,14)(13,15)(17,21,22,23)(18,27,28,25)(19,20,30,31)(24,32,29,26)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257394, #72257516, #72264103
maximal normal subgroup of	#72257394, #72257516, #72264103
maximal quotient of	#72308313

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6906123);