#120946

order	512
nilpotency class	3
isoclinism	isoclinic class of #120908
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1442
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

386 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=d,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,g^b=gh,e^c=ei,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28142);