#63090

order	256
nilpotency class	3
isoclinism	isoclinic class of #3585
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂³×D₄
commutator, Frattini, cosocle	C₄	C₂⁶
socle	C₂	C₂⁴×D₄
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

5418 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#63090⋊C₃, #72228335, #72258793
maximal subgroup of	#359543, #359548, #359575, …
maximal normal subgroup of	#359543, #359548, #359575, …
maximal quotient of	#10574061, #10574095, #10574166, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56078);