#493198

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₈	C₂⁴
commutator	C₂²	C₂²×C₄×C₈
socle	C₂³	C₂³×C₈
prime radical	C₂⁴	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

686 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)(13,14,17,23,18,16,28,30,19,22,31,37,32,29,21,36)(15,20,26,34,27,24,38,39,25,35,41,44,42,40,33,43);(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)(28,38)(29,40)(30,39)(31,41)(32,42)(36,43)(37,44);(13,16,17,30,18,22,28,37,19,29,31,36,32,14,21,23)(15,24,26,39,27,35,38,44,25,40,41,43,42,20,33,34);(5,7,8,11)(6,10,9,12)(14,22)(15,25)(16,29)(23,37)(26,41)(27,42)(30,36)(33,38)

Polycyclic presentation:

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

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400394);