#60398

order	256
nilpotency class	3
isoclinism	isoclinic class of #3026
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂×#120
commutator, Frattini, cosocle	C₂³	C₂⁵
socle	C₂	C₂²×#120
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ C₁

2418 conjugacy classes of subgroups; only the 557 normal subgroups are shown.

normal subgroups only

This group has 2418 conjugacy classes of subgroups — too many to draw. Only its 557 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#60398⋊C₃, #72233471, #72236910, …
maximal subgroup of	#147761, #147766, #364501, …
maximal normal subgroup of	#147761, #147766, #364501, …
maximal quotient of	#6346851, #6346852, #6346853, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53386);