#353823

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1678 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)(7,10)(8,9)(11,12)(13,14,17,23,19,24,18,22)(15,25,28,35,29,34,27,20)(16,21,31,38,30,36,32,37)(26,39,42,43,40,44,41,33);(13,15,18,27,19,29,17,28)(14,20,22,34,24,35,23,25)(16,26,32,41,30,40,31,42)(21,33,37,44,36,43,38,39);(6,9)(10,12)(13,16)(14,21)(15,26)(17,31)(18,32)(19,30)(20,33)(22,37)(23,38)(24,36)(25,39)(27,41)(28,42)(29,40)(34,44)(35,43);(5,7,8,11)(6,10,9,12)(16,30)(21,36)(26,40)(31,32)(33,43)(37,38)(39,44)(41,42)

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=f,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,e^b=ei,d^c=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261019);