#6395401

order	512
nilpotency class	3
isoclinism	isoclinic class of #61629
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

1742 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)(15,24)(16,21)(17,22)(18,20)(19,26)(23,31)(25,36)(27,34)(28,33)(29,37)(30,35)(32,40)(38,42)(39,44)(41,43);(1,2,4,3)(5,6)(7,10)(8,9)(11,12)(13,15,17,26)(14,19,22,24)(16,28,29,25)(18,27,31,40)(20,32,23,34)(21,36,37,33)(30,41,42,39)(35,44,38,43);(1,3,4,2)(14,20)(18,31)(19,32)(21,35)(22,23)(24,34)(27,40)(30,42)(33,43)(36,44)(37,38)(39,41);(6,9)(10,12)(13,16)(14,21)(15,25)(17,29)(18,30)(19,33)(20,35)(22,37)(23,38)(24,36)(26,28)(27,39)(31,42)(32,43)(34,44)(40,41);(5,7,8,11)(6,10,9,12)(14,22)(16,29)(19,24)(20,23)(25,28)(30,42)(32,34)(39,41)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6302597);