C₂×#13680 #512874

order	512
nilpotency class	5
isoclinism	isoclinic class of #13677
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13680

Subgroups

	subgroup	G/H
center, socle	C₂²	#2196
commutator, Frattini, cosocle	#124	C₂⁴
prime radical	C₂×#3289	C₂
LCS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×C₄ ◁ C₂×#124 ◁ G
LECS	G ▷ #124 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

1027 conjugacy classes of subgroups; only the 114 normal subgroups are shown.

normal subgroups only

This group has 1027 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);(3,4,10,15)(5,16,19,20)(6,17,11,23)(7,22,25,12)(8,13,27,32)(9,28,30,14)(18,34,33,24)(21,31,29,26);(3,5)(4,11)(6,15)(7,23)(8,26)(9,18)(10,19)(12,20)(13,24)(14,29)(16,22)(17,25)(21,28)(27,31)(30,33)(32,34);(3,6,10,11)(4,5,15,19)(7,24,25,34)(8,20,27,16)(9,29,30,21)(12,31,22,26)(13,17,32,23)(14,33,28,18)

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420070);