C₂×#10387 #238034

order	512
nilpotency class	3
isoclinism	isoclinic class of #10386
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#10387

Subgroups

	subgroup	G/H
center, socle	C₂³	#393
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1070 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1070 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) = 34
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 34 points:

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

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, 145230);