C₂×#8750 #190100

order	512
nilpotency class	3
isoclinism	isoclinic class of #8746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

direct product

C₂×#8750

Subgroups

	subgroup	G/H
center, socle	C₂³	#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

1518 conjugacy classes of subgroups; only the 242 normal subgroups are shown.

normal subgroups only

This group has 1518 conjugacy classes of subgroups — too many to draw. Only its 242 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)(5,11,13,8)(6,14)(7,10)(9,15)(12,16,18,17)(19,20)(21,24)(22,29,30,25)(23,26)(27,33,34,31)(28,32);(3,5)(4,8)(6,12)(7,13)(9,16)(10,11)(14,17)(15,18)(19,21,23,28)(20,24,26,32)(22,27,30,34)(25,31,29,33);(3,6)(4,9,10,14)(5,12)(7,15)(8,16,11,17)(13,18)(19,22)(20,25)(21,27)(23,30)(24,31)(26,29)(28,34)(32,33)

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97296);