#41740

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

690 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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=fg,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×#41740, #7639866, #7640241, …
maximal normal subgroup of	C₂×#41740, #7639866, #7640241, …
maximal quotient of	#544853, #545283, #545795, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34728);