#125426

order	512
nilpotency class	4
isoclinism	isoclinic class of #125425
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1555
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

304 conjugacy classes of subgroups; only the 66 normal subgroups are shown.

normal subgroups only

This group has 304 conjugacy classes of subgroups — too many to draw. Only its 66 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) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 32622);