#62969

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

Polycyclic presentation:

2^8|a^2=g,b^2=gh,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	#6350733, #6350734, #6350757, …
maximal normal subgroup of	#6350733, #6350734, #6350757, …
maximal quotient of	#7626631, #7626641, #7626646, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55957);