#6395455

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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 ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1737 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)(13,14,17,22)(15,24,25,19)(16,21,28,35)(18,20,30,23)(26,37,38,32)(27,31,39,33)(29,34,41,36)(40,42,44,43);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15)(14,19)(16,26)(17,25)(18,27)(20,33)(21,32)(22,24)(23,31)(28,38)(29,40)(30,39)(34,43)(35,37)(36,42)(41,44);(1,3,4,2)(14,20)(15,25)(18,30)(19,31)(21,34)(22,23)(24,33)(26,38)(29,41)(32,42)(35,36)(37,43);(6,9)(10,12)(13,16,17,28)(14,21,22,35)(15,26,25,38)(18,29,30,41)(19,32,24,37)(20,34,23,36)(27,40,39,44)(31,42,33,43);(5,7,8,11)(6,10,9,12)(14,22)(16,28)(19,24)(20,23)(26,38)(29,41)(31,33)(40,44)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6302651);