C₂×#7893 #180970

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁷

Aliases

direct product

C₂×#7893

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂⁴	C₂×C₂²⋊C₄
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2074 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 30 points:

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

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#121341, C₂×#182258, C₂×#116539, …
maximal normal subgroup of	C₂×#121341, C₂×#182258, C₂×#116539, …
maximal quotient of	C₂×#108795, C₂×#126846, #72321779, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88166);

C2×#7893 #180970

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#7893 #180970