#37522

order	256
nilpotency class	2
isoclinism	isoclinic class of #37436
automorphism	Aut(G) order 131072
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

731 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:

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

Polycyclic presentation:

2^8|a^2=g,b^2=f,c^2=f,e^2=g,b^a=bf,c^a=cg,d^a=df,e^a=efh,c^b=ch,d^c=dfh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6471266, #6471267, #6471278, …
maximal normal subgroup of	#6471266, #6471267, #6471278, …
maximal quotient of	#556345, #556383, #556411, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 30510);