#6393829

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹

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	C₂×#3554	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3098 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gh,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,e^c=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, 6301025);