C₂×#12290 #484950

order	512
nilpotency class	4
isoclinism	isoclinic class of #12102
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12290

Subgroups

	subgroup	G/H
center, socle	C₂³	#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#466	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂²×C₄⋊C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

804 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 804 conjugacy classes of subgroups — too many to draw. Only its 154 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);(3,4,7,10)(5,8,12,15)(6,9,13,16)(11,17,18,14)(20,25)(21,24)(22,29)(26,34)(27,33)(28,31)(30,32);(3,5)(4,8)(6,11)(7,12)(9,14)(10,15)(13,18)(16,17)(19,20)(21,26)(22,30)(23,27)(24,32)(25,29)(28,33)(31,34);(3,6,7,13)(4,9,10,16)(5,11,12,18)(8,14,15,17)(19,21,23,24)(20,26,27,32)(22,28,31,29)(25,30,33,34)

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, 392146);