#6400414

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×#548	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)(13,14,19,24)(15,20,28,36)(16,29,31,22)(17,23,18,21)(25,39,41,34)(26,35,27,33)(30,38,32,37)(40,44,42,43);(5,6)(7,10)(8,11)(9,12)(13,15,19,28)(14,20,24,36)(16,25,31,41)(17,26,18,27)(21,33,23,35)(22,34,29,39)(30,40,32,42)(37,43,38,44);(5,7,9,8)(6,10,12,11)(14,21)(17,18)(20,33)(22,37)(23,24)(26,27)(29,38)(30,32)(34,43)(35,36)(39,44)(40,42);(5,7,9,8)(6,11,12,10)(13,16,19,31)(14,22,24,29)(15,25,28,41)(17,32,18,30)(20,34,36,39)(21,37,23,38)(26,42,27,40)(33,43,35,44);(1,3,4,2)(5,8,9,7)(6,11,12,10)(13,17,19,18)(14,23,24,21)(15,26,28,27)(16,30,31,32)(20,35,36,33)(22,37,29,38)(25,40,41,42)(34,43,39,44)

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,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, 6307610);