#151152

order	512
nilpotency class	5
isoclinism	isoclinic class of #151150
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	#408	C₂³
prime radical	#23892	C₂
LCS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ #408 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 565 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

35 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58348);