#493306

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

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	C₂²×D₄	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

722 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)(7,11)(8,10)(9,12)(13,14,17,22,18,23,30,36,19,16,21,28,31,29,35,41)(15,20,25,32,26,33,38,42,24,27,34,39,37,40,43,44);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,27)(17,25)(18,26)(19,24)(21,34)(22,32)(23,33)(28,39)(29,40)(30,38)(31,37)(35,43)(36,42)(41,44);(7,12)(9,11)(13,16,17,28,18,29,30,41,19,14,21,22,31,23,35,36)(15,20,25,32,26,33,38,42,24,27,34,39,37,40,43,44);(6,10)(9,11)(15,24)(20,27)(25,34)(26,37)(32,39)(33,40)(38,43)(42,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=f,f^2=h,h^2=i,b^a=be,d^a=de,c^b=cei,d^b=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, 400502);