#6403933

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	C₂×#547	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1839 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)(7,11)(8,10)(9,12)(13,14,19,23)(15,20,26,36)(16,28,31,44)(17,22,18,33)(21,30,38,32)(24,39,29,43)(25,35,27,42)(34,40,37,41);(5,7)(6,9)(8,12)(10,11)(13,15,19,26)(14,20,23,36)(16,29,31,24)(17,25,18,27)(21,37,38,34)(22,35,33,42)(28,43,44,39)(30,41,32,40);(6,10)(9,11)(13,16,18,32,19,31,17,30)(14,21,22,28,23,38,33,44)(15,24,27,41,26,29,25,40)(20,34,35,39,36,37,42,43);(7,12)(9,11)(13,17,19,18)(14,22,23,33)(15,25,26,27)(16,30,31,32)(20,35,36,42)(21,28,38,44)(24,40,29,41)(34,39,37,43);(1,3,4,2)(6,10)(9,11)(15,26)(16,31)(20,36)(21,38)(25,27)(28,44)(30,32)(35,42)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fg,d^2=i,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,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, 6311129);