#41698

order	256
nilpotency class	2
isoclinism	isoclinic class of #40484
automorphism	Aut(G) = C₂¹⁵ (order 32768) · Inn(G) = C₂⁵ · Out(G) = C₂¹⁰
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

701 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

(2,5)(6,8)(9,10,14,18)(11,13,17,15)(12,16,22,24)(19,21,20,23)(25,26)(27,34)(28,32)(29,31)(30,33)(35,40)(36,39)(37,38);(11,17)(13,15)(19,20)(21,23)(25,27,31,28)(26,32,29,34)(30,36,38,35)(33,40,37,39);(1,2,4,5)(3,6,7,8)(9,11,14,17)(10,15,18,13)(12,20,22,19)(16,21,24,23)(25,28,31,27)(26,32,29,34)(30,36,38,35)(33,39,37,40);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,20)(18,24)(25,29,31,26)(27,34,28,32)(30,37,38,33)(35,40,36,39);(1,3,4,7)(2,6,5,8)(9,13,14,15)(10,17,18,11)(12,21,22,23)(16,20,24,19)(25,30)(26,33)(27,35)(28,36)(29,37)(31,38)(32,39)(34,40)

Polycyclic presentation:

2^8|a^2=h,b^2=f,c^2=fg,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#41698, #7639866, #7640237, …
maximal normal subgroup of	C₂×#41698, #7639866, #7640237, …
maximal quotient of	#544853, #545282, #545797, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34686);