C₂×#14663 #544369

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄²

Aliases

direct product

C₂×#14663

Subgroups

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

4914 conjugacy classes of subgroups; only the 1346 normal subgroups are shown.

normal subgroups only

This group has 4914 conjugacy classes of subgroups — too many to draw. Only its 1346 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);(11,12,14,16)(13,15,17,18)(20,24)(25,32)(26,29)(33,34);(4,6)(7,8,9,10)(13,17)(15,18)(19,20,23,24)(21,25,28,32)(22,29,31,26)(27,34,30,33);(3,4)(5,6)(7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21,23,28)(20,25,24,32)(22,30,31,27)(26,34,29,33);(11,13)(12,15)(14,17)(16,18)(19,22,23,31)(20,26,24,29)(21,27,28,30)(25,33,32,34)

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 451565);