#232237

order	512
nilpotency class	3
isoclinism	isoclinic class of #232235
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	C₂³×#41	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2406 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 2406 conjugacy classes of subgroups — too many to draw. Only its 218 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)(5,6)(7,9)(8,10)(11,12)(13,14,19,16)(15,22,23,28)(17,27,18,21)(20,25,26,24);(9,12)(13,15)(14,20)(16,26)(17,25)(18,24)(19,23)(21,28)(22,27);(2,4)(5,7)(6,9)(8,11)(10,12)(13,16,19,14)(15,22,23,28)(17,27,18,21)(20,24,26,25);(15,23)(20,26)(22,28)(24,25)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72278951
maximal normal subgroup of	#72278951

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139433);