#251974

order	512
nilpotency class	3
isoclinism	isoclinic class of #251967
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

713 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,d^2=i,f^2=h,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,d^c=di,e^c=eh,f^c=fh,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 159170);