C₂×#17229 #845050

order	512
nilpotency class	2
isoclinism	isoclinic class of #17082
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

direct product

C₂×#17229

Subgroups

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

3654 conjugacy classes of subgroups; only the 826 normal subgroups are shown.

normal subgroups only

This group has 3654 conjugacy classes of subgroups — too many to draw. Only its 826 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) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 30 points:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 752246);