C₂×#539088 #72325174

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#539088

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂⁵×D₄	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

40693 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:

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

Conjugacy classes & character table

232 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232168, C₂×#72232145, C₂×#72232159
maximal normal subgroup of	C₂×#72232168, C₂×#72232145, C₂×#72232159

GAP

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

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

C2×#539088 #72325174