#359172

order	512
nilpotency class	3
isoclinism	isoclinic class of #23444
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1968 conjugacy classes of subgroups; only the 306 normal subgroups are shown.

normal subgroups only

This group has 1968 conjugacy classes of subgroups — too many to draw. Only its 306 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) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

(1,2,4,3)(5,6)(7,12)(8,10)(9,11)(13,14)(15,18)(16,19)(17,20)(21,26)(22,25)(23,27)(24,28);(5,7,9,8)(6,10,11,12)(13,15)(14,18)(16,21)(17,22)(19,26)(20,25)(23,28)(24,27);(13,16)(14,19)(15,21)(17,23)(18,24)(20,27)(22,28)(25,26);(1,3,4,2)(5,8,9,7)(6,10,11,12)(16,23)(19,27)(21,28)(24,26)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266368);