C₂²×#2262 #7625187

order	512
nilpotency class	5
isoclinism	isoclinic class of D₃₂
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

direct product

C₂²×#2262

Subgroups

	subgroup	G/H
center	C₂²×C₄	D₁₆
commutator, Frattini, cosocle	C₁₆	C₂⁵
socle	C₂³	C₂×D₁₆
LCS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

1366 conjugacy classes of subgroups; only the 538 normal subgroups are shown.

normal subgroups only

This group has 1366 conjugacy classes of subgroups — too many to draw. Only its 538 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 — 5 generators on 68 points:

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

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, 7532383);