#125050

order	512
nilpotency class	4
isoclinism	isoclinic class of #124662
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#102
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ C₄²×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

388 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 98 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 — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fghi,d^2=gi,e^2=h,f^2=i,g^2=i,b^a=bd,c^a=cfgh,d^a=df,f^a=fi,g^a=gi,c^b=ci,d^b=dg,e^b=efgi,f^b=fi,g^b=gi

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, 32246);