C₂×#13855 #514023

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

direct product

C₂×#13855

Subgroups

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

3690 conjugacy classes of subgroups; only the 1410 normal subgroups are shown.

normal subgroups only

This group has 3690 conjugacy classes of subgroups — too many to draw. Only its 1410 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) = 512

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

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 421219);