C₂×#154115 #72334740

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

direct product

C₂×#154115

Subgroups

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

69597 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 26
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 26 points:

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

Conjugacy classes & character table

256 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72334389, C₂×#72334390, C₂×#72334398, …
maximal normal subgroup of	C₂×#72334389, C₂×#72334390, C₂×#72334398, …
maximal quotient of	#72337395

GAP

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

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

C2×#154115 #72334740