#61743

order	256
nilpotency class	3
isoclinism	isoclinic class of #61591
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 ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

885 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)(17,18,21,23)(19,27,29,32)(20,25,22,26)(24,30,31,28);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(18,23)(24,31)(25,26)(27,32);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,19,21,29)(18,24,23,31)(20,28,22,30)(25,32,26,27);(3,11)(6,9)(10,16)(13,15)(17,20,21,22)(18,25,23,26)(19,28,29,30)(24,32,31,27);(2,8)(6,9)(7,14)(13,15)(17,20,21,22)(18,26,23,25)(19,28,29,30)(24,27,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6362268, #6362272, #6362292, …
maximal normal subgroup of	#6362268, #6362272, #6362292, …
maximal quotient of	#6386165, #6387827, #6387912, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54731);