#393143

order	512
nilpotency class	3
isoclinism	isoclinic class of #32935
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂×C₄×D₄
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

918 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)(15,20,26,35,27,36,25,42)(16,28,32,38,30,44,31,21)(24,39,41,37,29,43,40,34);(6,9)(10,12)(13,15,18,26,19,27,17,25)(14,20,22,35,23,36,33,42)(16,29,32,40,30,24,31,41)(21,37,28,43,38,34,44,39);(13,16,17,31,19,30,18,32)(14,21,22,28,23,38,33,44)(15,24,25,40,27,29,26,41)(20,34,35,39,36,37,42,43);(2,4)(5,7,8,11)(6,10,9,12)(16,30)(21,38)(24,29)(28,44)(31,32)(34,37)(39,43)(40,41)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=f,d^2=e,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fi,c^b=ci,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, 300339);