#61762

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

878 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^8|c^2=fh,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×#61762, #10577936, #10578010, …
maximal normal subgroup of	C₂×#61762, #10577936, #10578010, …
maximal quotient of	#6386173, #6387836, #6387843, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54750);