#300486

order	512
nilpotency class	3
isoclinism	isoclinic class of #300469
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	#3561	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

820 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:

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

Polycyclic presentation:

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

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, 207682);