#232423

order	512
nilpotency class	3
isoclinism	isoclinic class of #232415
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1822 conjugacy classes of subgroups; only the 210 normal subgroups are shown.

normal subgroups only

This group has 1822 conjugacy classes of subgroups — too many to draw. Only its 210 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) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2)(3,4)(6,10)(7,8)(11,12)(13,14)(15,22)(16,27)(17,21)(18,20)(19,25)(23,24)(26,28);(5,6)(7,11)(8,10)(9,12)(13,15)(14,19)(16,26)(17,25)(18,24)(20,28)(21,23)(22,27);(2,4)(6,10)(7,8)(11,12)(13,14)(15,23)(16,21)(17,27)(18,20)(19,25)(22,24)(26,28);(5,7,9,8)(6,11,12,10)(14,20)(15,24)(21,27)(25,26)

Polycyclic presentation:

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

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, 139619);