#36039

order	256
nilpotency class	2
isoclinism	isoclinic class of #35680
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

769 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:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=f,c^2=f,d^2=fg,e^2=fg,b^a=bf,c^a=cg,c^b=ch,e^d=eh

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6586113, #6586125, #6586128, …
maximal normal subgroup of	#6586113, #6586125, #6586128, …
maximal quotient of	#517721, #517724, #517748, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29027);