#53042

order	256
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	Aut(G) order 65536
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

680 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:

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

Polycyclic presentation:

2^8|a^2=h,b^2=fg,c^2=f,d^2=g,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#7221601, #7221605, #7221665, …
maximal normal subgroup of	#7221601, #7221605, #7221665, …
maximal quotient of	#582294, #582493, #582534, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 46030);