#382463

order	512
nilpotency class	3
isoclinism	isoclinic class of #29634
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#379
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2228 conjugacy classes of subgroups; only the 356 normal subgroups are shown.

normal subgroups only

This group has 2228 conjugacy classes of subgroups — too many to draw. Only its 356 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=h,h^2=i,b^a=be,c^a=cf,g^a=gi,c^b=cg,f^b=fi,d^c=de

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 289659);