#149640

order	512
nilpotency class	5
isoclinism	isoclinic class of #13609
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#423	C₂³
socle	C₂	#11448
prime radical	C₂⁴	#101
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ #2905 ◁ G
LECS	G ▷ #423 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

243 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 243 conjugacy classes of subgroups — too many to draw. Only its 72 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=hi,e^2=g,f^2=i,g^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=ch,d^b=di,e^b=efh,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56836);