#60093

order	256
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	Aut(G) order 98304
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂³×D₄
commutator	C₄	C₂⁴×C₄
Frattini, cosocle	C₂×C₄	C₂⁵
socle	C₂	C₂³×C₂²⋊C₄
prime radical	C₂×#585	C₂
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

2426 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#60093⋊C₃, #72258723, #72258768
maximal subgroup of	#359649, #359652, #359653, …
maximal normal subgroup of	#359649, #359652, #359653, …
maximal quotient of	#6342908, #6342909, #6342910, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53081);