#150593

order	512
nilpotency class	5
isoclinism	isoclinic class of #150592
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#11798
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	#408	C₂³
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂²×C₈ ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #2923 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

257 conjugacy classes of subgroups; only the 50 normal subgroups are shown.

normal subgroups only

This group has 257 conjugacy classes of subgroups — too many to draw. Only its 50 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=h,d^2=gi,g^2=i,b^a=bd,c^a=cg,d^a=df,f^a=fh,h^a=hi,d^b=dg,e^b=efg,g^b=gi,d^c=di,e^c=ei,e^d=eh,f^e=fi

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, 57789);