#492566

order	512
nilpotency class	4
isoclinism	isoclinic class of #492504
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#19968	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

452 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2,5,7)(3,9,12,16)(4,13,14,8)(6,10,15,11)(17,18,22,27)(19,23,28,38)(20,24,33,41)(21,34,32,26)(25,42,40,31)(29,39,43,35)(30,47,44,37)(36,45,48,46);(1,3)(2,6)(4,10)(5,12)(7,15)(8,16)(9,13)(11,14)(17,19)(18,23)(20,29)(21,30)(22,28)(24,39)(25,36)(26,37)(27,38)(31,46)(32,44)(33,43)(34,47)(35,41)(40,48)(42,45);(18,24)(19,28)(20,31)(21,32)(23,35)(25,26)(27,41)(29,45)(33,42)(34,40)(36,47)(37,48)(38,39)(43,46);(2,7)(3,10)(4,14)(6,9)(11,12)(15,16)

Polycyclic presentation:

2^9|a^2=gi,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eg,f^a=fh,h^a=hi,c^b=ci,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399762);