#113315

order	512
nilpotency class	3
isoclinism	isoclinic class of #113311
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	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

732 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)(3,13,16,24,17,8,15,25)(4,18,21,27,22,9,20,28)(14,23,30,32,31,29,19,26)(33,34)(35,43,46,56)(36,47,49,40)(37,41)(38,42)(39,45,53,50)(44,58)(48,61,63,55)(51,57)(52,62)(54,59)(60,64);(1,3,7,17)(2,8,10,13)(4,19,22,30)(5,16,6,15)(9,26,18,32)(11,25,12,24)(14,21,31,20)(23,28,29,27)(33,35)(34,39)(36,44)(37,50)(38,46)(40,54)(41,56)(42,53)(43,57)(45,51)(47,52)(48,62)(49,60)(55,58)(59,63)(61,64);(1,4,7,22)(2,9,10,18)(3,14,17,31)(5,21,6,20)(8,23,13,29)(11,28,12,27)(15,30,16,19)(24,32,25,26)(33,36)(34,40)(35,44)(37,48)(38,49)(39,52)(41,55)(42,47)(43,58)(45,59)(46,60)(50,62)(51,63)(53,54)(56,64)(57,61)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,f^c=fh,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, 20511);