C₂²×#1657 #6351361

order	512
nilpotency class	3
isoclinism	isoclinic class of #1655
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#1657

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#120
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#496	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

3302 conjugacy classes of subgroups; only the 618 normal subgroups are shown.

normal subgroups only

This group has 3302 conjugacy classes of subgroups — too many to draw. Only its 618 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) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

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, 6258557);