#6671850

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#546	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2180 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

(6,9)(7,11)(13,14,17,19)(15,23,25,36)(16,27,28,21)(18,31,30,22)(20,32,33,26)(24,38,39,37)(29,35,43,42)(34,40,41,44);(1,2)(3,4)(5,6,8,9)(7,10,11,12)(14,19)(20,33)(21,27)(22,31)(23,36)(34,41)(35,42)(37,38);(2,4)(13,15,17,25)(14,20,19,33)(16,24,28,39)(18,32,30,26)(21,34,27,41)(22,36,31,23)(29,44,43,40)(35,38,42,37);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,28)(18,29)(19,27)(20,34)(22,35)(23,37)(25,39)(26,40)(30,43)(31,42)(32,44)(33,41)(36,38);(2,4)(6,9)(10,12)(16,28)(21,27)(24,39)(29,43)(34,41)(35,42)(37,38)(40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=i,d^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efhi,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6579046);