C₂×#62849 #10586004

order	512
nilpotency class	2
isoclinism	isoclinic class of #62842
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#62849

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂⁶
commutator, Frattini, cosocle	C₂²	C₂⁷
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂² ▷ C₁

32498 conjugacy classes of subgroups; only the 31674 normal subgroups are shown.

normal subgroups only

This group has 32498 conjugacy classes of subgroups — too many to draw. Only its 31674 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) = 128

A minimal faithful permutation representation — 7 generators on 42 points:

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

Conjugacy classes & character table

164 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#11710743
maximal subgroup of	C₂×#11710743, C₂×#6431839
maximal normal subgroup of	C₂×#11710743, C₂×#6431839

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10493200);