C₂×#17198 #844982

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄²

Aliases

direct product

C₂×#17198

Subgroups

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

4294 conjugacy classes of subgroups; only the 826 normal subgroups are shown.

normal subgroups only

This group has 4294 conjugacy classes of subgroups — too many to draw. Only its 826 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) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 30 points:

(1,2);(4,6)(7,8,10,12)(9,11,13,14)(15,16)(17,22)(18,21)(19,20)(23,24,26,28)(25,29,30,27);(9,13)(11,14)(16,20)(17,21)(23,24,26,28)(25,29,30,27);(7,9)(8,11)(10,13)(12,14)(15,17)(16,18)(19,21)(20,22)(23,24)(25,27)(26,28)(29,30);(3,4)(5,6)(7,8)(9,11)(10,12)(13,14)(15,18)(16,21)(17,20)(19,22)(23,25,26,30)(24,27,28,29)

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, 752178);