C₂×#32876 #6631466

order	512
nilpotency class	3
isoclinism	isoclinic class of #32875
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#32876

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1744 conjugacy classes of subgroups; only the 490 normal subgroups are shown.

normal subgroups only

This group has 1744 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);(3,4,8,12)(5,13,15,9)(6,17,19,10)(7,21,23,11)(14,29,30,24)(16,25,22,28)(18,26,33,31)(20,27,34,32);(3,5,7,22,8,15,23,16)(4,9,11,28,12,13,21,25)(6,18,20,14,19,33,34,30)(10,26,27,24,17,31,32,29);(3,6,7,20,8,19,23,34)(4,10,11,27,12,17,21,32)(5,14,16,18,15,30,22,33)(9,24,25,26,13,29,28,31);(14,30)(18,33)(24,29)(26,31)

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210064
maximal normal subgroup of	#72210064
maximal quotient of	C₂×#392035

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6538662);