#60064

order	256
nilpotency class	2
isoclinism	isoclinic class of #3484
automorphism	Aut(G) order 393216
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	C₂³×#41	C₂
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

2567 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 20
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 5 generators on 20 points:

(1,2)(3,4)(5,6,8,12,9,11,15,18)(7,10,14,17,13,16,19,20);(2,4);(5,7)(6,10)(8,14)(9,13)(11,16)(12,17)(15,19)(18,20);(7,13)(10,16)(14,19)(17,20);(6,11)(10,16)(12,18)(17,20)

Polycyclic presentation:

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

Conjugacy classes & character table

88 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#60064⋊C₃, #72228334, #72258735
maximal subgroup of	#105856, #105872, #105876, …
maximal normal subgroup of	#105856, #105872, #105876, …
maximal quotient of	#6342764, #6342765, #6342766, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53052);