#353747

order	512
nilpotency class	3
isoclinism	isoclinic class of #3057
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂³×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	C₂²×C₄⋊C₄
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

994 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260943);