#502850

order	512
nilpotency class	4
isoclinism	isoclinic class of #501040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

921 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)(8,9)(10,12)(13,14)(15,23)(16,19)(17,30)(18,31)(20,26)(21,24)(22,29)(25,35)(27,42)(28,43)(32,40)(33,39)(34,44)(36,38)(37,41);(1,2)(3,4)(6,9)(10,11)(14,19)(20,32)(21,33)(22,31)(23,35)(30,43)(34,36)(37,42);(2,4)(13,15,24,38,18,27,28,40,16,25,39,44,29,41,17,26)(14,20,30,37,22,34,33,35,19,32,43,42,31,36,21,23);(5,7,8,12)(6,10,9,11)(14,19)(20,32)(21,33)(22,31)(23,35)(30,43)(34,36)(37,42)

Polycyclic presentation:

2^9|c^2=fh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=cg

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410046);