#62894

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

3647 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

2^8|a^2=g,d^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	#62894⋊C₃, #72268133, #72268151, …
maximal subgroup of	#346551, #346559, #346567, …
maximal normal subgroup of	#346551, #346559, #346567, …
maximal quotient of	#7626570, #7626665, #7626666, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55882);