#392210

order	512
nilpotency class	3
isoclinism	isoclinic class of #184786
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×D₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

4474 conjugacy classes of subgroups; only the 442 normal subgroups are shown.

normal subgroups only

This group has 4474 conjugacy classes of subgroups — too many to draw. Only its 442 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)(3,4)(5,6)(7,10)(8,9)(11,12)(13,14)(15,18)(16,23,24,19)(17,20)(21,27,28,25)(22,26);(6,9)(10,12)(13,15)(14,18)(16,21)(17,22)(19,25)(20,26)(23,27)(24,28);(5,7,8,11)(6,10,9,12)(13,16)(14,19)(15,21)(17,24)(18,25)(20,23)(22,28)(26,27);(2,4)(15,22)(18,26)(21,28)(25,27)

Polycyclic presentation:

2^9|a^2=h,c^2=e,b^a=be,c^a=cf,d^a=dg,f^a=fi,d^b=di,h^c=hi

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, 299406);