#6400416

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

1846 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)(13,14,18,23)(15,25,28,20)(16,30,33,22)(17,24,19,21)(26,36,31,39)(27,35,29,37)(32,40,34,38)(41,43,42,44);(5,6)(7,10)(8,11)(9,12)(13,15,18,28)(14,20,23,25)(16,31,33,26)(17,27,19,29)(21,35,24,37)(22,39,30,36)(32,42,34,41)(38,44,40,43);(5,7,9,8)(6,10,12,11)(14,21)(17,19)(20,35)(22,38)(23,24)(25,37)(27,29)(30,40)(32,34)(36,43)(39,44)(41,42);(5,7,9,8)(6,11,12,10)(13,16,18,33)(14,22,23,30)(15,26,28,31)(17,34,19,32)(20,36,25,39)(21,38,24,40)(27,42,29,41)(35,43,37,44);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,17,18,19)(14,21,23,24)(15,27,28,29)(16,32,33,34)(20,35,25,37)(22,40,30,38)(26,41,31,42)(36,44,39,43)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,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, 6307612);