C₂×#17053 #612043

order	512
nilpotency class	2
isoclinism	isoclinic class of #16249
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

direct product

C₂×#17053

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁵	C₂⁴
commutator, Frattini, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3518 conjugacy classes of subgroups; only the 842 normal subgroups are shown.

normal subgroups only

This group has 3518 conjugacy classes of subgroups — too many to draw. Only its 842 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) = 512

A minimal faithful permutation representation — 5 generators on 34 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	C₂×#17053⋊C₃
maximal subgroup of	#72267734, #72267735, #72267738, …
maximal normal subgroup of	#72267734, #72267735, #72267738, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 519239);