#62893

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

3499 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 24 points:

(2,5)(6,8)(10,14)(12,20)(15,17)(18,24);(9,10,13,14)(11,17,19,15)(12,16,20,21)(18,23,24,22);(1,2)(3,6)(4,5)(7,8)(9,11,13,19)(10,15,14,17)(12,18,20,24)(16,22,21,23);(3,7)(6,8)(9,12)(10,16)(11,18)(13,20)(14,21)(15,22)(17,23)(19,24);(1,3)(2,6)(4,7)(5,8);(1,2)(3,6)(4,5)(7,8)

Polycyclic presentation:

2^8|b^2=g,c^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

Fitting subgroup of	#62893⋊C₃, #72268142, #72268160, …
maximal subgroup of	#346552, #346560, #346568, …
maximal normal subgroup of	#346552, #346560, #346568, …
maximal quotient of	#7626569, #7626571, #7626664, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55881);