C₂×#7351 #149555

order	512
nilpotency class	5
isoclinism	isoclinic class of #7349
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7351

Subgroups

	subgroup	G/H
center, socle	C₂²	#1317
commutator	C₂⁴	C₂²×C₈
Frattini, cosocle	#407	C₂³
prime radical	C₂⁵	C₂×C₈
LCS	G ▷ C₂⁴ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 436 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);(3,4,6,11,7,12,18,26,8,10,17,24,20,25,31,33)(5,9,14,21,15,23,28,32,13,22,16,29,19,27,30,34);(3,5)(4,9,10,22)(6,16)(7,19)(8,13)(11,21,24,29)(12,27,25,23)(14,17)(15,20)(18,28)(26,34,33,32)(30,31)

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, 56751);