C₂×#32933 #6629000

order	512
nilpotency class	3
isoclinism	isoclinic class of #32932
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

direct product

C₂×#32933

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

5102 conjugacy classes of subgroups; only the 802 normal subgroups are shown.

normal subgroups only

This group has 5102 conjugacy classes of subgroups — too many to draw. Only its 802 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 — 5 generators on 26 points:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287145, #72287176, #72287208
maximal normal subgroup of	#72287145, #72287176, #72287208
maximal quotient of	#72287773

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6536196);