C₂×#7433 #150280

order	512
nilpotency class	5
isoclinism	isoclinic class of #7396
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

direct product

C₂×#7433

Subgroups

	subgroup	G/H
center	C₂×C₄	#353
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#407	C₂³
socle	C₂²	#1352
prime radical	C₂⁵	Q₁₆
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#407 ◁ G
LECS	G ▷ #407 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

428 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 74 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);(3,4,7,11,8,13,15,12)(5,14,17,31,19,32,18,27)(6,20,22,28,24,30,23,10)(9,21,25,33,26,34,29,16);(3,5,15,17,8,19,7,18)(4,9,11,25,13,26,12,29)(6,21,22,33,24,34,23,16)(10,27,30,14,28,31,20,32)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57476);