#6685989

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682440
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 ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1888 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)(7,13)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22,25,30)(23,29,26,27)(24,32,35,36)(28,31,34,33);(1,2)(3,4)(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,26)(22,27,30,29)(24,33,35,31)(28,32,34,36);(2,4)(21,24,26,33,25,35,23,31)(22,28,29,32,30,34,27,36);(6,11)(10,13)(12,19)(18,20)(21,22)(23,27)(24,34)(25,30)(26,29)(28,35)(31,32)(33,36);(2,4)(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,23,25,26)(22,29,30,27)(24,31,35,33)(28,32,34,36)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=g,e^2=f,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^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283719, #72283845, #72284710
maximal normal subgroup of	#72283719, #72283845, #72284710

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593185);