#60401

order	256
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	Aut(G) order 49152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#120
commutator, Frattini, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×D₄ ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ C₁

2427 conjugacy classes of subgroups; only the 557 normal subgroups are shown.

normal subgroups only

This group has 2427 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#60401⋊C₃, #72236832, #72236841, …
maximal subgroup of	#133840, #499824, #499828, …
maximal normal subgroup of	#133840, #499824, #499828, …
maximal quotient of	#6352109, #6352110, #6352111, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53389);