#6400218

order	512
nilpotency class	3
isoclinism	isoclinic class of #61609
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

1844 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)(5,6,9,12)(7,11,8,10)(13,14,19,24)(15,20,28,35)(16,22,30,38)(17,23,18,21)(25,33,40,44)(26,34,27,32)(29,36,31,37)(39,42,41,43);(5,6)(7,10)(8,11)(9,12)(13,15,19,28)(14,20,24,35)(16,25,30,40)(17,26,18,27)(21,32,23,34)(22,33,38,44)(29,39,31,41)(36,42,37,43);(5,7,9,8)(6,10,12,11)(14,21)(17,18)(20,32)(22,36)(23,24)(26,27)(29,31)(33,42)(34,35)(37,38)(39,41)(43,44);(5,7,9,8)(6,11,12,10)(13,16,19,30)(14,22,24,38)(15,25,28,40)(17,31,18,29)(20,33,35,44)(21,36,23,37)(26,41,27,39)(32,42,34,43);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,23,24,21)(15,26,28,27)(16,29,30,31)(20,34,35,32)(22,36,38,37)(25,39,40,41)(33,42,44,43)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,d^2=fi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,e^c=ei,g^c=gi,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, 6307414);