#6675121

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667980
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3224	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1948 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(5,6)(7,15)(8,12)(9,14)(10,13)(11,17)(16,20)(18,19)(21,22)(23,30)(24,28)(25,27)(26,31)(29,33)(32,35)(34,36);(1,2)(3,4)(5,7,10,17)(6,11,13,15)(8,16,9,18)(12,19,14,20)(21,23)(22,26)(24,31)(25,32)(27,34)(28,30)(29,35)(33,36);(2,4)(5,8,10,9)(6,12,13,14)(7,16,17,18)(11,19,15,20)(22,27)(25,33)(26,34)(28,29)(30,35)(32,36);(6,13)(11,15)(12,14)(19,20)(22,28)(23,31)(27,29)(32,36);(2,4)(5,9,10,8)(6,14,13,12)(7,16,17,18)(11,19,15,20)(23,31)(26,30)(32,36)(34,35)

Polycyclic presentation:

2^9|b^2=fi,c^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,g^c=gi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308919, #72308985
maximal normal subgroup of	#72308919, #72308985

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6582317);