#249870

order	512
nilpotency class	3
isoclinism	isoclinic class of #249839
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3512	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

920 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)(3,12)(4,17)(5,14)(6,25)(7,10)(8,11)(9,21)(13,30)(15,24)(16,23)(18,29)(19,31)(20,28)(22,27)(26,32)(33,34,38,42)(35,39,44,54)(36,46,47,58)(37,49,52,41)(40,51,56,48)(43,59,60,63)(45,57,50,55)(53,62,64,61);(1,3)(2,5)(4,18)(6,15)(7,14)(8,16)(9,26)(10,23)(11,24)(12,25)(13,20)(17,19)(21,31)(22,32)(27,30)(28,29)(33,35,38,44)(34,39,42,54)(36,43,47,60)(37,50,52,45)(40,53,56,64)(41,57,49,55)(46,63,58,59)(48,62,51,61);(1,4,8,22)(2,9,11,28)(3,13,16,31)(5,19,24,30)(6,20,7,21)(10,27,25,17)(12,29,23,26)(14,18,15,32)(33,36)(34,40)(35,43)(37,51)(38,47)(39,53)(41,58)(42,56)(44,60)(45,62)(46,49)(48,52)(50,61)(54,64)(55,63)(57,59);(1,5)(2,3)(4,19)(6,23)(7,12)(8,24)(9,13)(10,15)(11,16)(14,25)(17,18)(20,26)(21,29)(22,30)(27,32)(28,31)(35,44)(37,52)(39,54)(41,49)(43,60)(46,58)(48,51)(53,64)

Polycyclic presentation:

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

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, 157066);