C₂×#14336 #515242

order	512
nilpotency class	2
isoclinism	isoclinic class of #2613
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹³

Aliases

direct product

C₂×#14336

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

18482 conjugacy classes of subgroups; only the 2114 normal subgroups are shown.

normal subgroups only

This group has 18482 conjugacy classes of subgroups — too many to draw. Only its 2114 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);(8,10)(15,16,20,18)(17,23,25,21)(19,22,27,26)(24,29,30,28);(12,14)(15,17,20,25)(16,21,18,23)(19,24,27,30)(22,28,26,29);(3,4,5,6)(7,8,9,10)(11,12,13,14)(15,18,20,16)(17,23,25,21)(19,22,27,26)(24,28,30,29);(15,19)(16,22)(17,24)(18,26)(20,27)(21,28)(23,29)(25,30)

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#515244, C₂×#517412, C₂×#616965
maximal normal subgroup of	C₂×#515244, C₂×#517412, C₂×#616965
maximal quotient of	#72340020, C₂×#154120

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 422438);