C₂×#7374 #149633

order	512
nilpotency class	5
isoclinism	isoclinic class of #1406
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₂×#7374

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂²	#1325
prime radical	C₂⁴	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#423 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

330 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 82 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 — 3 generators on 34 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56829);