C₂×#7350 #149553

order	512
nilpotency class	5
isoclinism	isoclinic class of #7348
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7350

Subgroups

	subgroup	G/H
center, socle	C₂²	#1317
commutator	C₂²×C₄	C₂²×C₈
Frattini, cosocle	#407	C₂³
prime radical	C₂⁵	M₁₆
LCS	G ▷ C₂²×C₄ ▷ 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);(3,4,7,12,8,11,21,27,9,13,20,30,6,18,19,29)(5,14,16,31,15,10,23,24,17,26,32,34,22,25,33,28);(3,5)(4,10,13,25)(6,15)(7,16)(8,22)(9,17)(11,14,18,26)(12,28,30,24)(19,23)(20,32)(21,33)(27,34,29,31)

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, 56749);