C₂×#13676 #512856

order	512
nilpotency class	5
isoclinism	isoclinic class of #13673
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13676

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	#127	C₂⁴
prime radical	C₂×#3291	C₂
LCS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#127 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1067 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 1067 conjugacy classes of subgroups — too many to draw. Only its 114 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) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#512863, #72279943, #72280012, …
maximal normal subgroup of	C₂×#512863, #72279943, #72280012, …
maximal quotient of	#72279943

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420052);