#392422

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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	C₂²×C₄⋊C₄
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1366 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)(7,9)(8,10)(11,12)(13,14,19,23,18,21,17,34)(15,24,28,35,27,39,26,20)(16,29,33,37,31,44,32,22)(25,38,42,43,30,36,41,40);(5,7,8,11)(6,9,10,12)(13,15,19,28,18,27,17,26)(14,20,34,39,21,35,23,24)(16,30,33,41,31,25,32,42)(22,38,44,40,37,36,29,43);(6,10)(9,12)(14,21)(16,31)(20,35)(23,34)(24,39)(25,30)(32,33)(41,42);(2,4)(13,16,17,32,18,31,19,33)(14,22,23,29,21,37,34,44)(15,25,26,41,27,30,28,42)(20,36,24,40,35,38,39,43)

Polycyclic presentation:

2^9|a^2=h,b^2=ef,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,g^a=gi,d^b=di,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, 299618);