#6810570

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×#449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1984 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fh,d^2=i,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,g^a=gi,c^b=ch,d^b=di,e^b=efh,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, 6717766);