C₂×#10390 #238070

order	512
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#10390

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2406 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 2406 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 26
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 26 points:

(1,2);(3,4)(5,7)(6,8)(9,10)(11,12)(13,16)(14,21,23,17)(15,18)(19,26,22,24)(20,25);(3,5)(4,7)(6,9)(8,10)(11,13)(12,16)(14,22,23,19)(15,20)(17,24,21,26)(18,25);(7,10)(11,14)(12,17)(13,19)(15,23)(16,24)(18,21)(20,22)(25,26)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#10390⋊C₃
maximal subgroup of	#72210088, #72240079, C₂×#346282, …
maximal normal subgroup of	#72210088, #72240079, C₂×#346282, …
maximal quotient of	C₂×#144629, C₂×#144631, C₂×#144633, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145266);