#392322

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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

1370 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)(13,14,18,22,19,23,17,33)(15,24,28,35,26,38,27,20)(16,29,31,36,32,43,30,21)(25,39,42,44,40,34,41,37);(6,9)(10,12)(13,15,17,27,19,26,18,28)(14,20,22,24,23,35,33,38)(16,25,30,41,32,40,31,42)(21,34,29,37,36,39,43,44);(5,7,8,11)(6,10,9,12)(13,16,17,30,19,32,18,31)(14,21,22,29,23,36,33,43)(15,25,27,41,26,40,28,42)(20,34,24,37,35,39,38,44);(2,4)(15,26)(20,35)(24,38)(25,40)(27,28)(34,39)(37,44)(41,42)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=ei,f^a=fi,d^b=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, 299518);