C₂×#17271 #845122

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

direct product

C₂×#17271

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

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, 752318);