C₂×#148104 #72309696

order	1024
nilpotency class	4
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

direct product

C₂×#148104

Subgroups

	subgroup	G/H
center, socle	C₂⁴	#442
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂⁴×C₄ ◁ C₂³×#97 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4829 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 256

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

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

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72249948, C₂×#72250110, C₂×#72249925
maximal normal subgroup of	C₂×#72249948, C₂×#72250110, C₂×#72249925

GAP

Past the SmallGroups library coverage — no SmallGroups id. Construct it from a minimal faithful permutation representation:

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

C2×#148104 #72309696