#317012

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	#28174	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 580 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)(3,10)(4,7)(5,8)(6,9)(11,14)(12,16)(13,15)(17,18)(19,27)(20,31)(21,34)(22,26)(23,29)(24,32)(25,35)(28,41)(30,39)(33,45)(36,44)(37,43)(38,46)(40,42)(47,48);(1,3,11,13,5,4,6,12)(2,7,14,16,8,10,9,15)(17,19)(18,23,26,36)(20,28)(21,29)(22,30)(24,37,38,41)(25,27,34,39)(31,40,45,47)(32,42)(33,43)(35,44)(46,48);(1,4,5,3)(2,7,8,10)(6,12,11,13)(9,16,14,15)(23,36)(27,39)(37,41)(40,47);(2,8)(7,10)(9,14)(15,16)(17,20,21,32,22,33,35,46)(18,24,25,31,26,38,34,45)(19,28,29,42,30,43,44,48)(23,37,27,40,36,41,39,47)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,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, 224208);