#233152

order	512
nilpotency class	3
isoclinism	isoclinic class of #233152
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2020 conjugacy classes of subgroups; only the 274 normal subgroups are shown.

normal subgroups only

This group has 2020 conjugacy classes of subgroups — too many to draw. Only its 274 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)(17,21)(20,24)(22,26)(23,25)(27,28);(2,4)(5,6)(7,11)(8,10)(9,12)(13,15)(14,18)(16,23)(17,22)(19,25)(20,27)(21,26)(24,28);(6,10)(7,8)(11,12)(14,19)(16,24)(18,25)(20,21)(23,28)(26,27);(15,22)(18,26)(23,28)(25,27)

Polycyclic presentation:

2^9|e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,g^b=gh,e^c=eh,f^c=fi,g^c=gh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140348);