#36057

order	256
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	Aut(G) order 98304
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	C₂⁵	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

829 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=fg,c^2=f,d^2=fg,e^2=f,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

Fitting subgroup of	#36057⋊C₃
maximal subgroup of	C₂×#36057, #7629527, #7629567, …
maximal normal subgroup of	C₂×#36057, #7629527, #7629567, …
maximal quotient of	#517956, #517964, #517969, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29045);