#6395308

order	512
nilpotency class	3
isoclinism	isoclinic class of #61643
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2283 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1748 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,g^a=gi,d^b=dfi,e^b=ei,g^c=gi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6302504);