C₂²×#1417 #512693

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂²×#1417

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₁₆
commutator	C₁₆	C₂³×C₄
Frattini, cosocle	C₂×C₁₆	C₂⁴
socle	C₂³	#357
prime radical	C₂²×#508	C₂
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

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

normal subgroups only

This group has 786 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);(5,6,9,14,11,17,30,16)(7,18,20,47,23,48,22,54)(8,24,27,55,29,57,62,56)(10,31,26,15,33,43,61,45)(12,34,35,50,37,49,46,19)(13,38,40,63,42,65,41,28)(21,52,51,66,53,67,32,39)(25,44,58,64,60,68,59,36);(5,7)(6,12)(8,25)(9,20)(10,32)(11,23)(13,39)(14,35)(15,44)(16,46)(17,37)(18,24)(19,28)(21,26)(22,30)(27,58)(29,60)(31,36)(33,51)(34,38)(40,52)(41,67)(42,66)(43,64)(45,68)(47,55)(48,57)(49,65)(50,63)(53,61)(54,56)(59,62)

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, 419889);