#6399558

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	C₂×#3269	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2209 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)(15,19)(16,21)(17,20)(18,23)(22,29)(24,31)(25,30)(26,33)(27,35)(28,34)(32,39)(36,40)(37,42)(38,41)(43,44);(6,9)(7,11)(13,15,18,26)(14,19,23,33)(16,24,27,37)(17,25,29,39)(20,30,22,32)(21,31,35,42)(28,38,40,44)(34,41,36,43);(14,20)(17,29)(19,30)(21,34)(22,23)(25,39)(28,40)(31,41)(32,33)(35,36)(38,44)(42,43);(5,7,8,11)(6,10,9,12)(13,16)(14,21)(15,24)(17,28)(18,27)(19,31)(20,34)(22,36)(23,35)(25,38)(26,37)(29,40)(30,41)(32,43)(33,42)(39,44);(1,3,4,2)(6,9)(10,12)(16,27)(21,35)(24,37)(28,40)(31,42)(34,36)(38,44)(41,43)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=f,e^2=h,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, 6306754);