C₂×#17273 #845124

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#17273

Subgroups

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

4546 conjugacy classes of subgroups; only the 826 normal subgroups are shown.

normal subgroups only

This group has 4546 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) = 34
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 34 points:

(1,2);(4,7)(8,10)(11,12,15,14)(13,16,17,18)(19,20,23,26)(21,22,25,24)(28,29)(30,34);(5,9)(8,10)(12,14)(13,17)(21,25)(22,24)(27,28,31,29)(30,32,34,33);(3,4,6,7)(5,8,9,10)(11,13)(12,16)(14,18)(15,17)(19,21,23,25)(20,24,26,22)(27,29,31,28)(30,32,34,33);(3,5,6,9)(4,8,7,10)(11,14,15,12)(13,16,17,18)(19,22,23,24)(20,25,26,21)(27,30)(28,32)(29,33)(31,34)

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, 752320);