#6677519

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1905 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308214, #72308293
maximal normal subgroup of	#72308214, #72308293

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6584715);