#120847

order	512
nilpotency class	3
isoclinism	isoclinic class of #120844
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1442
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#117
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

362 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 28043);