#300492

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	#3558	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

842 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:

(1,2,7,4)(3,11,14,26)(5,18,21,30)(6,17,19,10)(8,22,15,13)(9,24,16,20)(12,27,29,23)(25,31,32,28)(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);(1,3,6,13,7,14,19,22)(2,8,10,11,4,15,17,26)(5,12,20,28,21,29,24,31)(9,23,18,25,16,27,30,32)(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);(1,4,7,2)(3,8)(5,16,21,9)(6,17,19,10)(11,13)(12,23)(14,15)(18,20,30,24)(22,26)(25,28)(27,29)(31,32)(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);(1,5,19,24,7,21,6,20)(2,9,17,30,4,16,10,18)(3,12,22,31,14,29,13,28)(8,23,26,32,15,27,11,25)(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=fh,d^2=ehi,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, 207688);