C₂×#12306 #484981

order	512
nilpotency class	4
isoclinism	isoclinic class of #12304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

direct product

C₂×#12306

Subgroups

	subgroup	G/H
center, socle	C₂²	#1616
commutator, Frattini, cosocle	C₂×M₁₆	C₂⁴
prime radical	C₂×#469	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

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#486689, C₂×#485590, C₂×#486084, …
maximal normal subgroup of	C₂×#486689, C₂×#485590, C₂×#486084, …
maximal quotient of	#72307458, #72307573

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392177);