#63069

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

3248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

(1,2)(3,11)(4,14)(5,8)(6,10)(7,9)(12,13)(15,16)(17,18,23,22)(19,26,21,24)(20,25,30,29)(27,32,28,31);(1,3,8,7)(2,9,5,11)(4,12,10,15)(6,13,14,16)(17,19,23,21)(18,24,22,26)(20,28,30,27)(25,32,29,31);(1,4)(2,6)(3,12)(5,14)(7,15)(8,10)(9,13)(11,16)(17,20,23,30)(18,25,22,29)(19,27,21,28)(24,31,26,32);(1,5,8,2)(3,9,7,11)(4,14,10,6)(12,13,15,16)(17,21,23,19)(18,26,22,24)(20,28,30,27)(25,32,29,31);(1,6,8,14)(2,10,5,4)(3,13,7,16)(9,15,11,12)(17,18,23,22)(19,24,21,26)(20,25,30,29)(27,31,28,32);(1,7,8,3)(2,9,5,11)(4,15,10,12)(6,13,14,16)(17,22,23,18)(19,26,21,24)(20,29,30,25)(27,32,28,31)

Polycyclic presentation:

2^8|a^2=gh,b^2=g,c^2=gh,d^2=g,e^2=g,f^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	#63069⋊C₅
maximal subgroup of	#347449, #347469, #347496, …
maximal normal subgroup of	#347449, #347469, #347496, …
maximal quotient of	#7627422, #7627423, #7627537, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56057);