#299433

order	512
nilpotency class	3
isoclinism	isoclinic class of #299405
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

818 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

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, 206629);