#52988

order	256
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	Aut(G) order 393216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

1117 conjugacy classes of subgroups; only the 407 normal subgroups are shown.

normal subgroups only

This group has 1117 conjugacy classes of subgroups — too many to draw. Only its 407 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:

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

Polycyclic presentation:

2^8|c^2=fg,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#52988⋊C₃, #72340810, #72340817, …
maximal subgroup of	#132646, #132709, #487401, …
maximal normal subgroup of	#132646, #132709, #487401, …
maximal quotient of	#582225, #582442, #582608, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45976);