#6807215

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806551
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 ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1921 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)(7,13)(8,11)(9,12)(10,15)(14,19)(16,18)(17,20)(21,22,27,26)(23,28,32,35)(24,29,34,25)(30,33,36,31);(5,7)(6,10)(8,14)(9,15)(11,17)(12,13)(16,20)(18,19)(21,23)(22,28)(24,33)(25,30)(26,35)(27,32)(29,36)(31,34);(2,4)(5,8,9,16)(6,11,12,18)(7,14,15,20)(10,17,13,19)(21,24,26,25,27,34,22,29)(23,30,28,33,32,36,35,31);(6,12)(10,13)(11,18)(17,19)(21,22,27,26)(23,28,32,35)(24,29,34,25)(30,33,36,31);(7,15)(10,13)(14,20)(17,19)(21,25,26,34,27,29,22,24)(23,31,28,30,32,33,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308214, #72308783, #72308831
maximal normal subgroup of	#72308214, #72308783, #72308831

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714411);