C₂×#866702 #72337921

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#866702

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

36763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 30
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 30 points:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232336, C₂×#72232314, C₂×#72232346
maximal normal subgroup of	C₂×#72232336, C₂×#72232314, C₂×#72232346

GAP

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

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

C2×#866702 #72337921