C₂×#12465 #490556

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#12465

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#2213	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

450 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 138 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);(4,10)(5,15)(6,11)(7,22)(8,21)(13,33)(14,23)(16,17)(18,20)(19,27)(24,29)(25,26)(28,34)(30,31);(3,4,11,26,8,13,7,23,9,14,22,33,21,25,6,10)(5,12,27,29,17,30,16,28,19,32,15,34,18,31,20,24);(3,5)(4,12)(6,20)(7,16)(8,17)(9,19)(10,24)(11,27)(13,30)(14,32)(15,22)(18,21)(23,28)(25,31)(26,29)(33,34)

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397752);