#6404085

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2053 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)(5,6)(7,10)(8,9)(11,12)(13,14,18,22)(15,23,25,37)(16,20,27,34)(17,29,30,21)(19,31,33,26)(24,36,39,44)(28,41,40,35)(32,42,43,38);(6,9)(7,11);(13,15,17,26,18,25,30,31)(14,19,21,23,22,33,29,37)(16,24,28,38,27,39,40,42)(20,32,35,36,34,43,41,44);(5,7,8,11)(6,10,9,12)(13,16,18,27)(14,20,22,34)(15,24,25,39)(17,28,30,40)(19,32,33,43)(21,35,29,41)(23,36,37,44)(26,38,31,42);(1,3,4,2)(6,9)(10,12)(15,25)(16,27)(19,33)(20,34)(23,37)(26,31)(28,40)(35,41)

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^c=ei,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, 6311281);