#493309

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
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

708 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)(13,14,18,23,19,16,29,31,17,22,32,38,33,30,21,37)(15,24,27,40,28,41,39,44,26,20,34,36,43,25,35,42);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,25)(17,26)(18,27)(19,28)(21,35)(22,24)(23,36)(29,39)(30,41)(31,42)(32,34)(33,43)(37,44)(38,40);(7,12)(9,11)(13,16,18,31,19,22,29,38,17,30,32,37,33,14,21,23)(15,25,27,42,28,24,39,40,26,41,34,44,43,20,35,36);(6,10)(9,11)(15,26)(20,24)(25,41)(27,34)(28,43)(35,39)(36,40)(42,44)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400505);