#232453

order	512
nilpotency class	3
isoclinism	isoclinic class of #232451
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2180 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 2180 conjugacy classes of subgroups — too many to draw. Only its 204 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 — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,e^2=h,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gi,c^b=cg,d^b=dh,e^b=eh,g^b=gh,e^c=ei,f^c=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139649);