#149944

order	512
nilpotency class	5
isoclinism	isoclinic class of #149894
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2122
commutator	C₂×#41	C₂²×C₄
Frattini, cosocle	#408	C₂³
socle	C₂	#11557
prime radical	#24285	C₂
LCS	G ▷ C₂×#41 ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ #2929 ◁ G
LECS	G ▷ #408 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

403 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 403 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=g,e^2=g,f^2=i,g^2=i,b^a=bd,c^a=ci,d^a=df,f^a=fh,c^b=cg,e^b=efi,f^b=fi,g^b=gi,h^b=hi,e^d=ehi,f^d=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57140);