#61627

order	256
nilpotency class	3
isoclinism	isoclinic class of #61609
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #525 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

1025 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 40 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,16)(12,24)(13,18)(14,27)(15,19)(17,25)(20,35)(21,30)(22,31)(23,32)(26,33)(28,34)(29,36)(37,40)(38,39);(1,3)(2,5)(4,8)(6,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,23)(17,29)(18,30)(19,32)(24,35)(25,36)(26,38)(27,31)(28,37)(33,39)(34,40);(9,12,15,26)(10,17,19,34)(11,20,23,38)(13,25,14,28)(16,29,32,40)(18,33,27,24)(21,36,22,37)(30,39,31,35);(3,8)(5,7)(9,13,15,14)(10,18,19,27)(11,21,23,22)(12,25,26,28)(16,30,32,31)(17,33,34,24)(20,36,38,37)(29,39,40,35);(2,6)(5,7)(9,13,15,14)(10,18,19,27)(11,22,23,21)(12,25,26,28)(16,31,32,30)(17,33,34,24)(20,37,38,36)(29,35,40,39)

Polycyclic presentation:

2^8|c^2=h,d^2=h,e^2=h,g^2=h,b^a=bf,c^a=cg,e^a=ef,g^a=gh,d^b=df,e^b=eh,g^c=gh

Conjugacy classes & character table

49 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#61627, #10577230, #10577319, …
maximal normal subgroup of	C₂×#61627, #10577230, #10577319, …
maximal quotient of	#6386040, #6387628, #6387636, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54615);