C₂×#12308 #484986

order	512
nilpotency class	4
isoclinism	isoclinic class of #12307
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12308

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂⁴	AGL(1,ℤ/8)
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 696 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);(3,4,7,10,8,13,24,12)(5,14,16,29,17,32,23,31)(6,18,21,11,22,28,20,30)(9,19,26,15,27,33,25,34);(3,5,8,17)(4,9,13,27)(6,19,22,33)(7,23,24,16)(10,25,12,26)(11,29,30,31)(14,18,32,28)(15,21,34,20);(4,10,13,12)(9,25,27,26)(11,18,30,28)(14,29,32,31)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#486697, C₂×#486090, C₂×#485584
maximal normal subgroup of	C₂×#486697, C₂×#486090, C₂×#485584

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392182);