#6685803

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1926 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)(21,22,25,26)(23,27,31,29)(24,32,34,28)(30,36,33,35);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,25,31)(22,27,26,29)(24,33,34,30)(28,36,32,35);(1,2,3,4)(22,26)(23,29)(24,34)(27,31)(30,35)(33,36);(6,11)(10,13)(12,19)(18,20)(21,24,25,34)(22,28,26,32)(23,30,31,33)(27,35,29,36);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(22,26)(23,29)(27,31)(28,32)(30,36)(33,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250301, #72250457, #72256847, …
maximal normal subgroup of	#72250301, #72250457, #72256847, …
maximal quotient of	#72264204, #72264366, #72264407

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6592999);