#60400

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

2366 conjugacy classes of subgroups; only the 557 normal subgroups are shown.

normal subgroups only

This group has 2366 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:

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

Polycyclic presentation:

2^8|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	#11710347, #11710348, #11710349, …
maximal subgroup of	#133842, #499820, #499822, …
maximal normal subgroup of	#133842, #499820, #499822, …
maximal quotient of	#6352075, #6352076, #6352077, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 53388);