C₂×#7348 #149549

order	512
nilpotency class	5
isoclinism	isoclinic class of #7348
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#7348

Subgroups

	subgroup	G/H
center, socle	C₂²	#1317
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	#381	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂⁵ ◁ C₂×#381 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 486 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)(5,13,15,29,16,30,32,34)(6,17,18,27)(8,12,20,28)(9,21,22,31,23,33,26,14)(10,24,25,19);(3,5)(4,9,12,23)(6,14)(7,15)(8,16)(10,13,17,30)(11,26,28,22)(18,21)(19,31)(20,32)(24,33)(25,34,27,29)

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, 56745);