#62844

order	256
nilpotency class	2
isoclinism	isoclinic class of #62842
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

3737 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)(11,15)(12,19)(13,14)(16,20)(17,21)(18,24)(22,23);(1,2)(3,6)(4,5)(7,8)(11,17)(15,21)(18,23)(22,24);(1,3,4,7)(2,6,5,8)(10,14)(15,21)(16,19)(22,24);(9,10)(11,15)(12,16)(13,14)(17,21)(18,22)(19,20)(23,24);(9,11)(10,15)(12,18)(13,17)(14,21)(16,22)(19,24)(20,23);(9,12)(10,16)(11,18)(13,20)(14,19)(15,22)(17,23)(21,24)

Polycyclic presentation:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#62844⋊C₃, #72268136, #72268137, …
maximal subgroup of	#363969, #403943, #493458, …
maximal normal subgroup of	#363969, #403943, #493458, …
maximal quotient of	#7625682, #7625704, #7625705, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55832);