#63039

order	256
nilpotency class	2
isoclinism	isoclinic class of #63011
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

3560 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11710841, #11710842, #11710843
maximal subgroup of	#346943, #347197, #347206, …
maximal normal subgroup of	#346943, #347197, #347206, …
maximal quotient of	#7627233, #7627258, #7627259, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56027);