C₂×#7831 #165813

order	512
nilpotency class	3
isoclinism	isoclinic class of #7734
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

direct product

C₂×#7831

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

865 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 865 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 512

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

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 73009);