#374878

order	512
nilpotency class	3
isoclinism	isoclinic class of #23860
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#336 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1119 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=eh,d^2=f,e^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cg,e^b=ei,h^b=hi,e^d=ei,h^d=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282074);