#332624

order	512
nilpotency class	3
isoclinism	isoclinic class of #332583
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2668
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3497	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

675 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,5)(4,7)(6,8)(9,10,16,22)(11,23,27,42)(12,18,32,46)(13,33,36,49)(14,21,15,20)(17,28,41,26)(19,37,48,35)(24,52,54,43)(25,55,56,40)(29,59,58,39)(30,61,64,68)(31,50,38,45)(34,47,57,51)(44,63,69,62)(53,66,71,72)(60,70,65,67);(9,11,16,27)(10,17,22,41)(12,29,32,58)(13,34,36,57)(14,28,15,26)(18,43,46,52)(19,47,48,51)(20,42,21,23)(24,38,54,31)(25,37,56,35)(30,53,64,71)(33,40,49,55)(39,50,59,45)(44,66,69,72)(60,63,65,62)(61,67,68,70);(2,5)(3,8)(6,7)(9,12)(10,18)(11,24)(13,30)(14,31)(15,38)(16,32)(17,39)(19,44)(20,45)(21,50)(22,46)(23,43)(25,53)(26,29)(27,54)(28,58)(33,61)(34,65)(35,62)(36,64)(37,63)(40,66)(41,59)(42,52)(47,70)(48,69)(49,68)(51,67)(55,72)(56,71)(57,60);(2,5)(3,8)(6,7)(9,13,16,36)(10,19,22,48)(11,25,27,56)(12,30,32,64)(14,37,15,35)(17,40,41,55)(18,44,46,69)(20,49,21,33)(23,51,42,47)(24,53,54,71)(26,57,28,34)(29,60,58,65)(31,63,38,62)(39,66,59,72)(43,67,52,70)(45,68,50,61)

Polycyclic presentation:

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

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, 239820);