#6676442

order	512
nilpotency class	3
isoclinism	isoclinic class of #6668044
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	#30503	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1845 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=i,d^2=fi,e^2=f,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

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, 6583638);