C₂³×#497 #10575232

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³×C₄

Aliases

direct product

C₂³×#497

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁶
prime radical	C₂⁴×D₄	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

13242 conjugacy classes of subgroups; only the 4258 normal subgroups are shown.

normal subgroups only

This group has 13242 conjugacy classes of subgroups — too many to draw. Only its 4258 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) = 26
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10482428);