#61862

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
prime radical	#449	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ #533 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

807 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,9)(4,7)(5,8)(6,11)(10,15)(12,14)(13,16)(17,18,22,27)(19,29,31,24)(20,33,35,42)(21,28,23,26)(25,37,40,34)(30,44,45,48)(32,39,36,41)(38,47,43,46);(1,3,5,11)(2,6,8,9)(4,12,13,10)(7,15,16,14)(17,19,22,31)(18,24,27,29)(20,30,35,45)(21,36,23,32)(25,38,40,43)(26,41,28,39)(33,48,42,44)(34,47,37,46);(17,20)(18,25)(19,30)(21,34)(22,35)(23,37)(24,38)(26,33)(27,40)(28,42)(29,43)(31,45)(32,46)(36,47)(39,44)(41,48);(1,4,5,13)(2,7,8,16)(3,10,11,12)(6,14,9,15)(19,31)(24,29)(30,45)(32,36)(38,43)(39,41)(44,48)(46,47);(2,8)(6,9)(7,16)(14,15)(17,21,22,23)(18,26,27,28)(19,32,31,36)(20,34,35,37)(24,39,29,41)(25,33,40,42)(30,46,45,47)(38,44,43,48)

Polycyclic presentation:

2^8|a^2=h,b^2=f,d^2=fh,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×#61862, #10578259, #10578292, …
maximal normal subgroup of	C₂×#61862, #10578259, #10578292, …
maximal quotient of	#6386261, #6388022, #6388032, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54850);