#355229

order	512
nilpotency class	3
isoclinism	isoclinic class of #3192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂²×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	C₂×C₄×D₄
prime radical	C₂⁵	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂⁴×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1222 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)(16,28)(17,32)(18,22)(19,23)(21,30)(24,37)(25,41)(26,34)(27,35)(29,42)(31,36)(33,39)(38,44)(40,43);(5,7)(6,9)(8,12)(10,11)(13,15,18,26,19,27,17,25)(14,20,22,34,23,35,32,41)(16,24,31,40,29,38,30,39)(21,33,28,37,36,43,42,44);(7,12)(9,11)(13,16,17,30,19,29,18,31)(14,21,22,28,23,36,32,42)(15,24,25,39,27,38,26,40)(20,33,34,37,35,43,41,44);(6,10)(9,11)(16,29)(21,36)(24,38)(28,42)(30,31)(33,43)(37,44)(39,40)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,c^b=ce,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 262425);