#60402

order	256
nilpotency class	3
isoclinism	isoclinic class of #60399
automorphism	Aut(G) order 147456
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×Q₈ ◁ G
LECS	G ▷ C₂³ ▷ C₂ ▷ C₁

2389 conjugacy classes of subgroups; only the 557 normal subgroups are shown.

normal subgroups only

This group has 2389 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:

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

Polycyclic presentation:

2^8|a^2=h,d^2=h,e^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	#11710351, #11710352, #11710353, …
maximal subgroup of	#133841, #499823, #499827, …
maximal normal subgroup of	#133841, #499823, #499827, …
maximal quotient of	#6352123, #6352124, #6352125, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53390);