#232301

order	512
nilpotency class	3
isoclinism	isoclinic class of #232291
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35001	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2000 conjugacy classes of subgroups; only the 266 normal subgroups are shown.

normal subgroups only

This group has 2000 conjugacy classes of subgroups — too many to draw. Only its 266 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)(15,18)(16,19)(17,20)(21,24)(22,25)(23,27)(26,28);(5,6)(7,11)(8,10)(9,12)(13,15,17,22)(14,18,20,25)(16,21,23,28)(19,26,27,24);(2,4)(6,10)(7,8)(11,12)(13,16)(14,19)(15,21)(17,23)(18,24)(20,27)(22,28)(25,26);(5,7,9,8)(6,10,12,11)

Polycyclic presentation:

2^9|b^2=hi,d^2=h,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=dh,e^b=eh,g^b=gh,d^c=dh,e^c=ei,g^c=gh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139497);