C₄×#1420 #151967

order	512
nilpotency class	5
isoclinism	isoclinic class of #1418
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

direct product

C₄×#1420

Subgroups

	subgroup	G/H
center	C₂×C₄	#357
commutator	C₁₆	C₂×C₄²
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#357
prime radical	C₂³	C₂×SD₃₂
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

286 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 286 conjugacy classes of subgroups — too many to draw. Only its 116 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 — 3 generators on 68 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59163);