C₂×#8804 #191310

order	512
nilpotency class	3
isoclinism	isoclinic class of #8803
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

direct product

C₂×#8804

Subgroups

	subgroup	G/H
center, socle	C₂³	#383
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#126
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 488 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) = 128

A minimal faithful permutation representation — 4 generators on 34 points:

(1,2);(3,4)(5,12,14,16)(6,8)(7,11)(9,13,17,18)(10,15)(19,20)(21,29,31,34)(22,26,32,25)(23,27)(24,28)(30,33);(4,8,11,10)(9,12,17,16)(19,21,30,32,24,31,23,22)(20,25,27,29,28,26,33,34);(3,5,6,13,7,14,15,18)(4,9,10,16,11,17,8,12)(19,22,23,31,24,32,30,21)(20,26,27,34,28,25,33,29)

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 98506);