#496268

order	512
nilpotency class	4
isoclinism	isoclinic class of #33758
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁵	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1110 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cei,f^c=fh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 403464);