#241656

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

757 conjugacy classes of subgroups; only the 140 normal subgroups are shown.

normal subgroups only

This group has 757 conjugacy classes of subgroups — too many to draw. Only its 140 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) = 256

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

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,d^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 148852);