#62652

order	256
nilpotency class	2
isoclinism	isoclinic class of H(7,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂²	C₂⁶
commutator	C₂	C₂⁵×C₄
prime radical	C₂²×H(5,𝔽₂)	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

5837 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 20 points:

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

Polycyclic presentation:

2^8|a^2=h,b^a=bg,f^a=fg,e^b=eg,d^c=dg

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11710601, #11710602, #72241364, …
maximal subgroup of	#392237, #392239, #392240, …
maximal normal subgroup of	#392237, #392239, #392240, …
maximal quotient of	#7625218, #7625220, #7625221, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55640);