#62197

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 ◁ #529 ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

851 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=f,c^2=h,d^2=h,e^2=fh,g^2=h,b^a=bf,c^a=cg,d^a=dh,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×#62197, #10579450, #10579522, …
maximal normal subgroup of	C₂×#62197, #10579450, #10579522, …
maximal quotient of	#6386449, #6388559, #6388639, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55185);