#6686160

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667484
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×#121 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1932 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=gi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dfi,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308000, #72308130, #72308535
maximal normal subgroup of	#72308000, #72308130, #72308535

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6593356);