#61532

order	256
nilpotency class	3
isoclinism	isoclinic class of #61071
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 ◁ #529 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

891 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

55 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#61532, #10576867, #10576929, …
maximal normal subgroup of	C₂×#61532, #10576867, #10576929, …
maximal quotient of	#6379588, #6379595, #6380434, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54520);