#62694

order	256
nilpotency class	2
isoclinism	isoclinic class of #560
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	C₂⁴
commutator, Frattini, socle, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄² ◁ G
LECS	G ▷ C₂² ▷ C₁

3320 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^8|e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11710658, #72234950, #72234951, …
Fitting subgroup of	#11710658, #11710659, #11710660, …
maximal subgroup of	#187578, #187581, #187596, …
maximal normal subgroup of	#187578, #187581, #187596, …
maximal quotient of	#7625931, C₄×#3529, C₄×#3531, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55682);