#47825

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

672 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:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=g,c^2=f,d^2=f,e^2=h,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×#47825, #7701900, #7701902, …
maximal normal subgroup of	C₂×#47825, #7701900, #7701902, …
maximal quotient of	#576703, #576764, #577197, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 40813);