C₂×#13611 #512672

order	512
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#13611

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₄⋊Q₈	C₂⁴
prime radical	C₂×#3221	C₂
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂³×C₄ ◁ C₂×#536 ◁ G
LECS	G ▷ C₄⋊Q₈ ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

824 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 186 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);(3,4,6,12,8,13,19,27)(5,10,15,23,14,24,28,34)(7,9,17,22,20,11,18,26)(16,21,31,32,29,25,30,33);(4,9)(5,14)(6,17)(7,20)(10,21)(11,13)(15,30)(18,19)(22,26)(24,25)(28,31)(32,33);(3,5)(4,10)(6,15)(7,16)(8,14)(9,21)(11,25)(12,23)(13,24)(17,31)(18,30)(19,28)(20,29)(22,32)(26,33)(27,34)

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#152139, C₂×#512676, C₂×#152143
maximal normal subgroup of	C₂×#152139, C₂×#512676, C₂×#152143
maximal quotient of	#72293102, #72293104, #72293200

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419868);