#124862

order	512
nilpotency class	4
isoclinism	isoclinic class of #124861
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1558
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

394 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 394 conjugacy classes of subgroups — too many to draw. Only its 78 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=g,c^2=i,d^2=gi,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cfh,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efg,f^b=fi,h^b=hi,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, 32058);