C₂×#1393576 #72338009

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

Aliases

direct product

C₂×#1393576

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂×#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

37605 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:

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

Conjugacy classes & character table

172 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232287, C₂×#72232297, C₂×#72232330
maximal normal subgroup of	C₂×#72232287, C₂×#72232297, C₂×#72232330

GAP

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

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

C2×#1393576 #72338009