#60065

order	256
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂×C₄	C₂⁵
commutator, socle	C₂²	C₂⁴×C₄
prime radical	C₂×#585	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

2437 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

(1,2)(3,4)(5,6,9,13,10,14,20,26)(7,11,17,23,16,22,29,33)(8,12,19,25,18,24,30,34)(15,21,28,32,27,31,35,36);(2,4)(5,7)(6,11)(8,15)(9,17)(10,16)(12,21)(13,23)(14,22)(18,27)(19,28)(20,29)(24,31)(25,32)(26,33)(30,35)(34,36);(5,8)(6,12)(7,15)(9,19)(10,18)(11,21)(13,25)(14,24)(16,27)(17,28)(20,30)(22,31)(23,32)(26,34)(29,35)(33,36);(8,18)(12,24)(15,27)(19,30)(21,31)(25,34)(28,35)(32,36);(7,16)(11,22)(15,27)(17,29)(21,31)(23,33)(28,35)(32,36)

Polycyclic presentation:

2^8|a^2=g,g^2=h,b^a=bf,e^b=eh,d^c=dh

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#60065⋊C₃
maximal subgroup of	#129204, #129205, #129228, …
maximal normal subgroup of	#129204, #129205, #129228, …
maximal quotient of	#6342836, #6342837, #6342838, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53053);