#151133

order	512
nilpotency class	5
isoclinism	isoclinic class of #151132
automorphism	?
homology	Abelianization (H₁) = C₂³ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#12090
commutator, Frattini, cosocle	#381	C₂³
prime radical	#23896	C₂
LCS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³ ◁ C₂⁴ ◁ #381 ◁ G
LECS	G ▷ #381 ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁

593 conjugacy classes of subgroups; only the 38 normal subgroups are shown.

normal subgroups only

This group has 593 conjugacy classes of subgroups — too many to draw. Only its 38 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) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

(1,2,7,12)(3,13,17,42)(4,18,21,38)(5,10,24,37)(6,25,28,11)(8,29,32,15)(9,26,34,20)(14,43,46,63)(16,48,50,41)(19,53,22,52)(23,39,27,36)(30,59,60,64)(31,47,58,57)(33,56,35,55)(40,49,62,45)(44,51,54,61);(1,3)(2,8)(4,14)(5,15)(6,16)(7,17)(9,30)(10,13)(11,31)(12,32)(18,51)(19,54)(20,45)(21,46)(22,44)(23,57)(24,29)(25,58)(26,49)(27,47)(28,50)(33,62)(34,60)(35,40)(36,48)(37,42)(38,61)(39,41)(43,53)(52,63)(55,64)(56,59);(1,4)(2,9)(3,14)(5,22)(6,26)(7,21)(8,30)(10,35)(11,38)(12,34)(13,40)(15,44)(16,49)(17,46)(18,25)(19,24)(20,28)(23,55)(27,56)(29,54)(31,61)(32,60)(33,37)(36,53)(39,52)(41,63)(42,62)(43,48)(45,50)(47,59)(51,58)(57,64)

Polycyclic presentation:

2^9|a^2=i,d^2=f,e^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,g^a=gh,d^b=df,e^b=egh,g^b=gi,h^b=hi,d^c=dg,e^c=ei,f^c=fi,e^d=ehi,g^d=gi

Conjugacy classes & character table

32 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285485
maximal normal subgroup of	#72285485

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58329);