#46996

order	256
nilpotency class	2
isoclinism	isoclinic class of #46350
automorphism	Aut(G) order 786432
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1429 conjugacy classes of subgroups; only the 411 normal subgroups are shown.

normal subgroups only

This group has 1429 conjugacy classes of subgroups — too many to draw. Only its 411 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) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 24 points:

(1,2)(3,7)(4,5)(6,8)(9,10)(11,14)(12,13)(15,16);(1,2,4,5)(3,7,8,6)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,20,23,24);(2,5)(6,7)(10,13)(14,15)(18,22)(19,23);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,20)(21,23)(22,24);(1,3)(2,6)(4,8)(5,7)(9,11)(10,14)(12,16)(13,15)(17,20)(18,19)(21,24)(22,23)

Polycyclic presentation:

2^8|b^2=fgh,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#46996⋊C₃, #72231789, #72231804, …
maximal subgroup of	#130062, #133422, #251827, …
maximal normal subgroup of	#130062, #133422, #251827, …
maximal quotient of	#576607, #577215, #577914, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39984);