C₂×#32874 #6631459

order	512
nilpotency class	3
isoclinism	isoclinic class of #32873
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄²

Aliases

direct product

C₂×#32874

Subgroups

	subgroup	G/H
center, socle	C₂²	#2840
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2054 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 2054 conjugacy classes of subgroups — too many to draw. Only its 490 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) = 64

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

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#144642, C₂×#6631461, C₂×#7266802, …
maximal normal subgroup of	C₂×#144642, C₂×#6631461, C₂×#7266802, …
maximal quotient of	#72235846

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538655);