#504936

order	512
nilpotency class	4
isoclinism	isoclinic class of #499712
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

507 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,d^2=ef,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^b=gh,h^b=hi,g^c=gh,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412132);