#6807281

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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 ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2001 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|b^2=i,c^2=fhi,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,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256939, #72256954, #72257109
maximal normal subgroup of	#72256939, #72256954, #72257109

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6714477);