#510025

order	512
nilpotency class	4
isoclinism	isoclinic class of #509081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

574 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417221);