#114624

order	512
nilpotency class	3
isoclinism	isoclinic class of #114623
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

760 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)(5,18)(6,9)(8,17)(10,20)(12,27,28,32)(13,22,31,26)(16,25)(19,24)(21,29,23,30)(33,34)(35,44,48,59)(36,41)(37,42)(38,53)(39,43)(40,46,55,47)(45,61,63,56)(49,64,62,54)(50,57)(51,60)(52,58);(1,3)(2,7)(4,12)(5,13)(6,14)(8,23)(9,11)(10,22)(15,27)(16,28)(17,29)(18,26)(19,30)(20,31)(21,24)(25,32)(33,35)(34,40)(36,49)(37,47)(38,46)(39,48)(41,56)(42,59)(43,55)(44,53)(45,51)(50,63)(52,62)(54,57)(58,61)(60,64);(1,4)(2,8)(3,12)(5,19)(6,16)(7,21)(9,24)(10,25)(11,23)(13,30)(14,28)(15,18)(17,20)(22,27)(26,32)(29,31)(33,36,39,52)(34,41,43,58)(35,45,48,63)(37,51,38,50)(40,54,55,64)(42,60,53,57)(44,61,59,56)(46,49,47,62)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,e^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fhi,c^b=cf,d^b=di,f^b=fi,g^b=ghi,d^c=dhi,e^c=eh,f^c=fi

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, 21820);