#124772

order	512
nilpotency class	4
isoclinism	isoclinic class of #124718
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

307 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 307 conjugacy classes of subgroups — too many to draw. Only its 70 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) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,c^2=fh,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=ch,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di,e^c=ei

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, 31968);