#62966

order	256
nilpotency class	2
isoclinism	isoclinic class of #62892
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

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

3332 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:

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

Polycyclic presentation:

2^8|b^2=h,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=cg,e^d=eh

Conjugacy classes & character table

73 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6350833, #6350834, #6350857, …
maximal normal subgroup of	#6350833, #6350834, #6350857, …
maximal quotient of	#7626630, #7626640, #7626650, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55954);