#6623997

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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	#24196	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1643 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=gi,d^2=fgh,e^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257305, #72257541, #72264159
maximal normal subgroup of	#72257305, #72257541, #72264159
maximal quotient of	#72264500

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6531193);