C₂×#10612 #242360

order	512
nilpotency class	3
isoclinism	isoclinic class of #10611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#10612

Subgroups

	subgroup	G/H
center, socle	C₂³	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 848 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);(3,4,6,12,8,10,7,11)(5,13,14,9,16,17,15,18)(19,20)(21,29)(22,27)(23,26)(24,28)(25,31)(30,33)(32,34);(3,5,7,15,8,16,6,14)(4,9,12,17,10,18,11,13)(19,21,23,32,24,30,22,31)(20,25,27,33,28,34,26,29);(4,10)(5,14)(6,7)(9,17)(13,18)(15,16)(20,26)(21,30)(22,23)(25,29)(27,28)(33,34)

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, 149556);