#113109

order	512
nilpotency class	3
isoclinism	isoclinic class of #113096
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

748 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)(3,13,16,23,17,8,15,24)(4,9,19,27,20,26,21,28)(14,29,18,32,30,22,31,25)(33,34)(35,44)(36,41)(37,53)(38,42)(39,43)(40,46)(45,56)(47,59)(48,55)(49,64)(50,60)(51,57)(52,58)(54,62)(61,63);(1,3,7,17)(2,8,10,13)(4,18,20,31)(5,16,6,15)(9,25,26,32)(11,24,12,23)(14,21,30,19)(22,28,29,27)(33,35,39,47)(34,40,43,55)(36,49,52,62)(37,48,38,46)(41,56,58,61)(42,59,53,44)(45,50,63,51)(54,60,64,57);(1,4,7,20)(2,9,10,26)(3,14,17,30)(5,21,6,19)(8,22,13,29)(11,28,12,27)(15,31,16,18)(23,32,24,25)(33,36,39,52)(34,41,43,58)(35,45,47,63)(37,51,38,50)(40,54,55,64)(42,60,53,57)(44,61,59,56)(46,49,48,62)

Polycyclic presentation:

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

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, 20305);