#150901

order	512
nilpotency class	5
isoclinism	isoclinic class of #150413
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2128
commutator	C₂×M₁₆	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂	#11899
prime radical	#532	D₄
LCS	G ▷ C₂×M₁₆ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2931 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

245 conjugacy classes of subgroups; only the 60 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 60 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=fh,c^2=gh,d^2=gh,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=cfi,d^b=dg,e^b=efghi,f^b=fi,h^b=hi,d^c=dhi,e^c=ei,f^c=fi,e^d=eh,f^d=fi,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 58097);