C₂×#759464 #72323922

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴×C₄

Aliases

direct product

C₂×#759464

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁴×#121	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

71017 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:

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

Conjugacy classes & character table

208 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72237111, C₂×#72323271, C₂×#72323264, …
maximal normal subgroup of	C₂×#72237111, C₂×#72323271, C₂×#72323264, …
maximal quotient of	C₂×#72334305, C₂×#72298468

GAP

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

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

C2×#759464 #72323922