C₂×#606717 #72294561

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

direct product

C₂×#606717

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator	C₂³	C₂⁵×C₄
Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂²×#2847	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

31901 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:

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

Conjugacy classes & character table

184 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72293912
maximal normal subgroup of	C₂×#72293912
maximal quotient of	C₂×#72293960

GAP

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

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

C2×#606717 #72294561