#6809523

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
prime radical	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1835 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)(8,12)(9,11)(10,15)(14,20)(16,18)(17,19)(21,22)(23,32)(24,34)(25,29)(26,30)(27,31)(28,33)(35,36);(5,7,9,15)(6,10,11,13)(8,16,17,14)(12,20,19,18)(21,23,26,33)(22,28,30,32)(24,27,25,35)(29,31,34,36);(2,4)(21,24,27,33,26,25,35,23)(22,29,31,32,30,34,36,28);(6,11)(10,13)(12,19)(18,20)(22,30)(28,32)(29,34)(31,36);(5,8,9,17)(6,12,11,19)(7,14,15,16)(10,18,13,20)(21,25,35,33,26,24,27,23)(22,29,36,28,30,34,31,32)

Polycyclic presentation:

2^9|b^2=f,c^2=h,e^2=fh,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	#72283745, #72284169, #72284411, …
maximal normal subgroup of	#72283745, #72284169, #72284411, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6716719);