C₂×#12174 #484655

order	512
nilpotency class	4
isoclinism	isoclinic class of #12173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#12174

Subgroups

	subgroup	G/H
center, socle	C₂²	#1598
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×M₁₆ ◁ G
LECS	G ▷ C₂×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 896 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)(5,15)(6,20)(7,11)(8,13)(9,14)(10,16)(12,22)(17,28)(18,31)(19,32)(21,25)(23,29)(24,30)(26,34)(27,33);(3,5,9,19)(4,10,14,27)(6,21,23,34)(7,18,8,17)(11,25,13,26)(12,28,30,31)(15,29,32,20)(16,24,33,22);(4,11,14,13)(10,25,27,26)(12,29,30,20)(15,28,32,31)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#486341, C₂×#486090, C₂×#485584
maximal normal subgroup of	C₂×#486341, C₂×#486090, C₂×#485584

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391851);