C₂×#918357 #72337879

order	1024
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²×C₄

Aliases

direct product

C₂×#918357

Subgroups

	subgroup	G/H
center, socle	C₂⁵	C₂⁵
commutator, Frattini, cosocle	C₂⁴	C₂⁶
prime radical	C₂³×#392	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

48849 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:

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

Conjugacy classes & character table

160 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#72232360, C₂×#72232334, C₂×#72232299
maximal normal subgroup of	C₂×#72232360, C₂×#72232334, C₂×#72232299

GAP

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

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

C2×#918357 #72337879