#233473

order	512
nilpotency class	3
isoclinism	isoclinic class of #233469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1688 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1688 conjugacy classes of subgroups — too many to draw. Only its 222 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)(7,10)(8,11)(9,12)(13,14)(15,23)(16,19)(17,21)(18,22)(20,24)(25,28)(26,27);(2,4)(6,10)(7,8)(11,12)(14,19)(16,17)(20,27)(21,22)(23,28)(24,25);(5,6)(7,10)(8,11)(9,12)(13,15)(14,20)(16,24)(17,25)(18,26)(19,27)(21,23)(22,28);(5,7,9,8)(6,10,12,11)(13,16,18,17)(14,21,22,19)(15,24,26,25)(20,23,28,27)

Polycyclic presentation:

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

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, 140669);