#63091

order	256
nilpotency class	3
isoclinism	isoclinic class of #63091
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂⁴×D₄
commutator, Frattini, cosocle	C₄	C₂⁶
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ #585 ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

5447 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

82 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11710897, #11710898, #63091⋊C₅, …
maximal subgroup of	#511763, #511765, #511767, …
maximal normal subgroup of	#511763, #511765, #511767, …
maximal quotient of	#10574063, #10574098, #10574172, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56079);