#114600

order	512
nilpotency class	3
isoclinism	isoclinic class of #114587
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

724 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 21796);