#151156

order	512
nilpotency class	5
isoclinism	isoclinic class of #151154
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

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

normal subgroups only

This group has 562 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)(3,15,19,51)(4,20,25,49)(5,26,31,12)(6,32,7,13)(9,36,41,16)(10,28,45,17)(11,34,50,24)(18,56,33,46)(21,40,22,39)(23,61,29,58)(27,43,55,44)(30,52,35,53)(37,64,48,63)(38,54,47,62)(42,60,57,59);(1,3)(2,9)(4,21)(5,27)(6,33)(7,18)(8,19)(10,37)(11,46)(12,42)(13,40)(14,41)(15,20)(16,24)(17,35)(22,25)(23,62)(26,57)(28,30)(29,54)(31,55)(32,39)(34,36)(38,52)(43,58)(44,61)(45,48)(47,53)(49,51)(50,56)(59,63)(60,64);(1,3)(2,10)(4,22)(5,28)(6,18)(7,33)(8,19)(9,37)(11,47)(12,51)(13,43)(14,45)(15,26)(16,24)(17,31)(20,57)(21,25)(23,59)(27,30)(29,60)(32,44)(34,36)(35,55)(38,50)(39,61)(40,58)(41,48)(42,49)(46,53)(52,56)(54,64)(62,63)

Polycyclic presentation:

2^9|a^2=i,d^2=fi,e^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=df,e^a=ei,f^a=fi,g^a=gh,d^b=df,e^b=eghi,f^b=fi,h^b=hi,d^c=dg,e^c=ei,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, 58352);