C₂×#148095 #72322871

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

Aliases

direct product

C₂×#148095

Subgroups

	subgroup	G/H
center, socle	C₂³	#1879
commutator	C₂²×C₈	C₂³×C₄
Frattini, cosocle	C₂³×C₈	C₂⁴
prime radical	C₂³×#135	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ C₂²×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

4545 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 38
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 38 points:

(7,8,10,17)(9,18,19,13)(11,14,22,31)(12,26,23,16)(15,24,30,25)(20,32,35,27)(21,29,36,34)(28,38,33,37);(1,2);(3,4)(5,6)(7,9,10,19)(8,13)(11,20,22,35)(12,21,23,36)(14,27)(15,28)(16,29)(17,18)(24,37,25,38)(26,34)(30,33)(31,32);(4,6)(8,14,17,31)(11,24)(12,23)(13,27,18,32)(15,26,30,16)(20,37)(21,36)(22,25)(28,34,33,29)(35,38)

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72249887, C₂×#72249861
maximal normal subgroup of	C₂×#72249887, C₂×#72249861

GAP

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

gap> G := Group((7,8,10,17)(9,18,19,13)(11,14,22,31)(12,26,23,16)(15,24,30,25)(20,32,35,27)(21,29,36,34)(28,38,33,37), (1,2), (3,4)(5,6)(7,9,10,19)(8,13)(11,20,22,35)(12,21,23,36)(14,27)(15,28)(16,29)(17,18)(24,37,25,38)(26,34)(30,33)(31,32), (4,6)(8,14,17,31)(11,24)(12,23)(13,27,18,32)(15,26,30,16)(20,37)(21,36)(22,25)(28,34,33,29)(35,38));

C2×#148095 #72322871