C₂×#636488 #72339987

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄²

Aliases

direct product

C₂×#636488

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#127	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

80989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 26 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72339988, C₂×#72339897
maximal normal subgroup of	#72339988, C₂×#72339897
maximal quotient of	C₂×#72339923

GAP

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

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

C2×#636488 #72339987