#61908

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×Q₈ ◁ G
LECS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁

867 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)(3,11)(4,8)(5,9)(6,10)(7,13)(12,16)(14,15)(17,18,21,26)(19,25,22,23)(20,28,29,31)(24,30,32,27);(1,3,6,13)(2,7,10,11)(4,12,5,14)(8,15,9,16)(17,19,21,22)(18,23,26,25)(20,27,29,30)(24,28,32,31);(1,4,6,5)(2,8,10,9)(3,12,13,14)(7,15,11,16)(17,20,21,29)(18,24,26,32)(19,27,22,30)(23,28,25,31);(1,5,6,4)(2,9,10,8)(3,12,13,14)(7,15,11,16)(17,19,21,22)(18,25,26,23)(20,27,29,30)(24,31,32,28);(2,10)(7,11)(8,9)(15,16)(17,19,21,22)(18,23,26,25)(20,27,29,30)(24,28,32,31)

Polycyclic presentation:

2^8|a^2=h,b^2=f,c^2=f,d^2=f,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	#6362269, #6362273, #6362293, …
maximal normal subgroup of	#6362269, #6362273, #6362293, …
maximal quotient of	#6386323, #6388211, #6388213, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 54896);