#37367

order	256
nilpotency class	2
isoclinism	isoclinic class of #36702
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	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

730 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)(9,10,13,12)(11,15,16,14)(17,18)(19,26)(20,21)(22,30)(23,25)(24,31)(27,29)(28,32);(2,5)(6,8)(17,19,23,20)(18,21,25,26)(22,27,31,28)(24,29,30,32);(1,2)(3,6)(4,5)(7,8)(9,11,13,16)(10,14,12,15)(17,20,23,19)(18,21,25,26)(22,28,31,27)(24,29,30,32);(1,2)(3,6)(4,5)(7,8)(9,12,13,10)(11,15,16,14)(17,21)(18,19)(20,25)(22,29)(23,26)(24,27)(28,30)(31,32);(1,3,4,7)(2,6,5,8)(17,22,23,31)(18,24,25,30)(19,27,20,28)(21,29,26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6486324, #6486331, #6486343, …
maximal normal subgroup of	#6486324, #6486331, #6486343, …
maximal quotient of	#553823, #553836, #554053, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 30355);