C₂×#7339 #149529

order	512
nilpotency class	5
isoclinism	isoclinic class of #7339
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#7339

Subgroups

	subgroup	G/H
center, socle	C₂²	#1316
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	#381	C₂³
prime radical	C₂³×#41	C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁵ ◁ C₂×#381 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

1110 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 1110 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);(3,4,7,11)(5,13,15,31,16,26,32,34)(6,17,19,29)(8,12,21,30)(9,22,23,18,24,33,28,14)(10,25,27,20);(3,5)(4,9)(6,18)(7,15)(8,16)(10,26)(11,28)(12,24)(13,17)(14,20)(19,33)(21,32)(22,25)(23,30)(27,31)(29,34)

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, 56725);