#149868

order	512
nilpotency class	5
isoclinism	isoclinic class of #149792
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2126
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#423	C₂³
socle	C₂	#11532
prime radical	C₂³	#413
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ #2960 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

225 conjugacy classes of subgroups; only the 56 normal subgroups are shown.

normal subgroups only

This group has 225 conjugacy classes of subgroups — too many to draw. Only its 56 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=gh,d^2=hi,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,c^b=ci,d^b=dh,e^b=efh,f^b=fi,e^c=ei,e^d=ehi,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57064);