#6687689

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682564
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 ◁ C₄²×D₄ ◁ 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307538, #72307631, #72308479
maximal normal subgroup of	#72307538, #72307631, #72308479

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6594885);