C₂×#7904149 #72299006

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂¹⁹

Aliases

direct product

C₂×#7904149

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂⁶
commutator, Frattini, cosocle	C₂³	C₂⁷
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂³ ▷ C₁

At least 6572 conjugacy classes of subgroups; only the 31853 normal subgroups are shown.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 30 points:

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

Conjugacy classes & character table

190 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72297863
maximal normal subgroup of	C₂×#72297863
maximal quotient of	C₂×#72298084

GAP

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

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

C2×#7904149 #72299006