#6404192

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×#548	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2066 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)(15,20)(16,28)(17,22)(18,34)(19,24)(21,31)(23,26)(25,38)(27,36)(29,43)(30,44)(32,35)(33,37)(39,42)(40,41);(6,9)(7,11)(13,15,19,27)(14,20,24,36)(16,25,30,41)(17,26,33,18)(21,35,39,43)(22,23,37,34)(28,38,44,40)(29,31,32,42);(13,16,26,42,19,30,18,31)(14,21,34,44,24,39,23,28)(15,25,33,29,27,41,17,32)(20,35,22,40,36,43,37,38);(5,7,8,11)(6,10,9,12)(13,17)(14,22)(15,26)(16,29)(18,27)(19,33)(20,23)(21,38)(24,37)(25,31)(28,43)(30,32)(34,36)(35,44)(39,40)(41,42);(1,3,4,2)(6,9)(10,12)(15,27)(16,30)(17,33)(20,36)(21,39)(22,37)(28,44)(31,42)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=gi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^b=ei,d^c=di,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, 6311388);