#63045

order	256
nilpotency class	2
isoclinism	isoclinic class of #63041
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

3742 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#63045⋊C₃, #72234215, #72268540, …
maximal subgroup of	#347434, #347454, #347474, …
maximal normal subgroup of	#347434, #347454, #347474, …
maximal quotient of	#7627404, #7627406, #7627424, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56033);