C₂×#14349 #515264

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#14349

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁴
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

5586 conjugacy classes of subgroups; only the 1346 normal subgroups are shown.

normal subgroups only

This group has 5586 conjugacy classes of subgroups — too many to draw. Only its 1346 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);(7,8,11,9)(10,12,14,13)(16,19)(20,21)(23,24)(25,28)(26,29)(27,30);(8,9)(12,13)(15,16,18,19)(17,20,22,21)(23,25)(24,27)(26,30)(28,29);(3,4,5,6)(7,9,11,8)(10,12,14,13)(15,16)(17,21)(18,19)(20,22)(23,25)(24,28)(26,30)(27,29);(7,10)(8,12)(9,13)(11,14)(15,17)(16,20)(18,22)(19,21)(23,24)(25,27)(26,29)(28,30)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422460);