#317007

order	512
nilpotency class	3
isoclinism	isoclinic class of #316991
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

553 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 553 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224203);