C₂×#13594 #512622

order	512
nilpotency class	5
isoclinism	isoclinic class of #1412
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#13594

Subgroups

	subgroup	G/H
center	C₂×C₄	#354
commutator	C₄²	C₂³×C₄
Frattini, cosocle	#127	C₂⁴
socle	C₂²	C₂×#354
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#532 ◁ G
LECS	G ▷ #127 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁

848 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 848 conjugacy classes of subgroups — too many to draw. Only its 194 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)(5,14,17,24)(6,10)(7,12,21,11)(8,13)(9,16,23,20)(15,28,30,34)(18,26,32,25)(19,27)(22,29,33,31);(3,5,8,16)(4,9)(6,15,19,29)(7,20,21,17)(10,22)(11,14)(12,23)(13,24)(18,31,32,30)(25,28)(26,33)(27,34);(3,6,8,19)(4,10,13,27)(5,15,16,29)(7,18,21,32)(9,22,24,34)(11,25,12,26)(14,28,23,33)(17,30,20,31)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419818);