#113084

order	512
nilpotency class	3
isoclinism	isoclinic class of #113076
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

786 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:

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

Polycyclic presentation:

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

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, 20280);