#62962

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

3283 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:

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

Polycyclic presentation:

2^8|a^2=g,b^2=gh,c^2=g,d^2=gh,e^2=h,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	#346801, #346825, #346849, …
maximal normal subgroup of	#346801, #346825, #346849, …
maximal quotient of	#7626626, #7626635, #7626710, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55950);