C₂×#12309 #484987

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

Aliases

direct product

C₂×#12309

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

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

normal subgroups only

This group has 712 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);(3,4,9,13,8,14,7,11)(5,15,17,32,19,33,18,29)(6,20,23,30,22,28,24,12)(10,25,26,16,27,21,31,34);(3,5)(4,10)(6,21)(7,17)(8,19)(9,18)(11,26)(12,29)(13,31)(14,27)(15,28)(16,23)(20,33)(22,25)(24,34)(30,32);(4,11,14,13)(6,22)(10,26,27,31)(12,28,30,20)(15,32,33,29)(16,34)(21,25)(23,24)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#486701, C₂×#486084, C₂×#485583, …
maximal normal subgroup of	C₂×#486701, C₂×#486084, C₂×#485583, …
maximal quotient of	#72265439, #72307573

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 392183);