#493316

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

716 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)(13,14,17,22,18,16,28,29,19,23,31,37,32,30,21,36)(15,20,26,35,27,24,38,40,25,34,41,44,42,39,33,43);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(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);(7,12)(9,11)(13,16,17,29,18,23,28,37,19,30,31,36,32,14,21,22)(15,24,26,40,27,34,38,44,25,39,41,43,42,20,33,35);(6,10)(9,11)(15,25)(20,34)(24,39)(26,41)(27,42)(33,38)(35,44)(40,43)

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=de,c^b=ce,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, 400512);