C₂×#7994 #181280

order	512
nilpotency class	3
isoclinism	isoclinic class of #1284
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

direct product

C₂×#7994

Subgroups

	subgroup	G/H
center	C₂²×C₄	#95
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#95
prime radical	C₂⁵	C₄⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1752 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 26 points:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#202149, C₂×#181290
maximal normal subgroup of	C₂×#202149, C₂×#181290
maximal quotient of	#72309112

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 88476);

C2×#7994 #181280

Aliases

Subgroups

Presentation

Conjugacy classes & character table

Relationships

GAP

C₂×#7994 #181280