#62999

order	256
nilpotency class	2
isoclinism	isoclinic class of D₄×H(5,𝔽₂)
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

3557 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

85 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11710808, #11710809, #11710810
maximal subgroup of	#347246, #347270, #347294, …
maximal normal subgroup of	#347246, #347270, #347294, …
maximal quotient of	#7625786, #7625816, #7625821, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55987);