#63011

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

4404 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 16
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 6 generators on 16 points:

(1,2)(3,6)(4,7)(5,8)(9,10)(11,14)(12,15)(13,16);(1,2)(3,6)(4,7)(5,8)(9,11)(10,13)(12,16)(14,15);(1,3)(2,5)(4,8)(6,7)(9,11)(10,13)(12,16)(14,15);(9,11)(10,13)(12,16)(14,15);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15);(1,2)(3,5)(4,7)(6,8)(9,10)(11,13)(12,15)(14,16)

Polycyclic presentation:

2^8|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	#63011⋊C₃, #72235205, #72235300
maximal subgroup of	#346934, #347190, #493635, …
maximal normal subgroup of	#346934, #347190, #493635, …
maximal quotient of	#7627211, #7627216, #7627266, …
automorphism group of	#344, #421

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55999);