C₂×#7851 #180851

order	512
nilpotency class	2
isoclinism	isoclinic class of #392
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#7851

Subgroups

	subgroup	G/H
center	C₂²×C₄²	C₂³
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂⁴	C₂×C₄²
prime radical	C₂⁵	C₄²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1718 conjugacy classes of subgroups; only the 738 normal subgroups are shown.

normal subgroups only

This group has 1718 conjugacy classes of subgroups — too many to draw. Only its 738 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) = 38
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 38 points:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88047);