C₂×#7318 #128027

order	512
nilpotency class	4
isoclinism	isoclinic class of #7318
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#7318

Subgroups

	subgroup	G/H
center, socle	C₂²	#1304
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂³ ▷ C₂ ▷ C₁

1438 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 1438 conjugacy classes of subgroups — too many to draw. Only its 146 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 — 3 generators on 34 points:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72225899, C₂×#128028, #72258332, …
maximal normal subgroup of	#72225899, C₂×#128028, #72258332, …
maximal quotient of	#72225888, C₂×#94453, #72226362, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35223);