#232280

order	512
nilpotency class	3
isoclinism	isoclinic class of #232273
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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

1898 conjugacy classes of subgroups; only the 218 normal subgroups are shown.

normal subgroups only

This group has 1898 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)(6,10)(7,8)(11,12)(13,14)(15,24)(16,21)(17,28)(18,22)(19,23)(20,25)(26,27);(5,6,9,12)(7,11,8,10)(13,15)(14,20)(16,23)(17,21)(18,25)(19,26)(22,27)(24,28);(2,4)(6,10)(7,8)(11,12)(13,16)(14,21)(15,22)(17,27)(18,24)(19,20)(23,25)(26,28);(5,7,9,8)(6,11,12,10)(13,17,19,18)(14,22,23,28)(15,21,26,25)(16,24,20,27)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 139476);