C₂×#10213 #232853

order	512
nilpotency class	3
isoclinism	isoclinic class of #10212
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#10213

Subgroups

	subgroup	G/H
center, socle	C₂³	#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3253	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1360 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 1360 conjugacy classes of subgroups — too many to draw. Only its 202 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) = 128

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140049);