C₂×#30236 #6435886

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#30236

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1176 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 1176 conjugacy classes of subgroups — too many to draw. Only its 490 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343082);