C₂×#13712 #512977

order	512
nilpotency class	5
isoclinism	isoclinic class of #13661
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#13712

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×D₁₆
commutator, Frattini, cosocle	C₂×C₁₆	C₂⁴
prime radical	C₂×#2215	C₂
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆ ◁ G
LECS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

426 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 426 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 42
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 42 points:

(1,2);(3,4)(5,8)(6,7)(9,10)(11,12)(13,17)(14,18)(15,28)(16,29)(19,25)(20,26)(21,24)(22,30)(23,40)(27,36)(31,37)(32,38)(33,41)(34,39)(35,42);(4,7)(8,10)(12,17)(18,30)(19,31)(20,32)(21,29)(28,40)(33,35)(34,36);(3,5,6,9)(4,8,7,10)(12,18)(14,25)(15,26)(16,24)(17,30)(19,20)(21,36)(22,37)(23,38)(27,42)(28,33)(29,34)(31,32)(35,40)(39,41)

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#513003, #72322860, #72322878
maximal normal subgroup of	C₂×#513003, #72322860, #72322878
maximal quotient of	#72321810, #72321831

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 420173);