#46109

order	256
nilpotency class	2
isoclinism	isoclinic class of #45438
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	C₂×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

790 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=f,b^2=f,c^2=f,d^2=fgh,e^2=g,b^a=bf,c^a=cg,d^a=dh,e^a=ef,c^b=ch,d^b=dfh,e^b=eh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#7221580, #7221584, #7221644, …
maximal normal subgroup of	#7221580, #7221584, #7221644, …
maximal quotient of	#557279, #557342, #557486, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39097);