C₂²×#1425 #512717

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂²×#1425

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂³	#366
prime radical	C₂⁴	Q₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

422 conjugacy classes of subgroups; only the 282 normal subgroups are shown.

normal subgroups only

This group has 422 conjugacy classes of subgroups — too many to draw. Only its 282 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) = 68
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 68 points:

(1,2);(3,4);(5,6,9,15,12,17,11,33)(7,18,21,43,24,45,23,55)(8,25,27,56,29,58,28,59)(10,31,30,41,32,60,26,16)(13,34,35,51,37,50,61,20)(14,38,39,48,40,47,57,19)(22,53,52,65,54,68,49,44)(36,46,42,62,63,67,64,66);(5,7,19,46,10,22,27,34,9,21,38,62,30,52,29,51,12,24,48,67,32,54,28,50,11,23,47,66,26,49,8,20)(6,13,25,44,16,36,57,55,33,61,59,68,60,64,40,45,17,37,58,65,41,63,39,43,15,35,56,53,31,42,14,18)

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419913);