#60399

order	256
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	?
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

2564 conjugacy classes of subgroups; only the 557 normal subgroups are shown.

normal subgroups only

This group has 2564 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) = 16
transitive degree	μ_t(G) = 16

A minimal faithful permutation representation — 5 generators on 16 points:

(6,12)(13,16);(1,2)(3,6)(4,7)(5,8)(9,13)(10,12)(11,14)(15,16);(1,3)(2,6)(4,9)(5,10)(7,13)(8,12)(11,15)(14,16);(1,4)(2,7)(3,9)(5,11)(6,13)(8,14)(10,15)(12,16);(4,11)(7,14)(9,15)(13,16)

Polycyclic presentation:

2^8|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	#60399⋊C₃, #72236801, #72236803, …
maximal subgroup of	#133839, #499819, #499821, …
maximal normal subgroup of	#133839, #499819, #499821, …
maximal quotient of	#6352061, #6352062, #6352063, …
outer automorphism group of	#1963, #1995

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53387);