#47432

order	256
nilpotency class	2
isoclinism	isoclinic class of #46350
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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

674 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)(9,10,14,12)(11,17,18,15)(13,16,21,20)(19,23,24,22)(25,26,30,28)(27,33,34,31)(29,36,37,32)(35,38,40,39);(1,2,5,4)(3,6,7,8)(11,18)(13,21)(15,17)(16,20)(26,28)(27,34)(29,37)(38,39);(1,3)(2,6)(4,8)(5,7)(9,11)(10,15)(12,17)(13,19)(14,18)(16,22)(20,23)(21,24)(25,27,30,34)(26,31,28,33)(29,35,37,40)(32,38,36,39);(1,2,5,4)(3,6,7,8)(9,12,14,10)(11,15,18,17)(13,20,21,16)(19,22,24,23)(25,28,30,26)(27,31,34,33)(29,36,37,32)(35,38,40,39);(1,4,5,2)(3,8,7,6)(9,13,14,21)(10,16,12,20)(11,19,18,24)(15,22,17,23)(25,29)(26,32)(27,35)(28,36)(30,37)(31,38)(33,39)(34,40)

Polycyclic presentation:

2^8|a^2=h,b^2=gh,c^2=f,d^2=g,e^2=fg,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#47432, #7701341, #7701392, …
maximal normal subgroup of	C₂×#47432, #7701341, #7701392, …
maximal quotient of	#576925, #577010, #577138, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 40420);