#249997

order	512
nilpotency class	3
isoclinism	isoclinic class of #249943
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	C₂³×C₄	C₂⁴
prime radical	#525	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

951 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)(3,12)(4,7)(5,20)(6,19)(8,11)(9,14)(10,21)(13,24)(15,26)(16,17)(18,23)(22,25)(27,28)(29,31)(30,32)(33,34,38,42)(35,39,45,52)(36,40,47,54)(37,48,49,41)(43,55,57,50)(44,58,59,51)(46,53,60,61)(56,63,64,62);(1,3,7,15)(2,9,4,17)(5,13,22,29)(6,16,8,14)(10,24,18,31)(11,26,19,12)(20,27,25,32)(21,30,23,28)(33,35)(34,39)(36,43)(37,44)(38,45)(40,50)(41,51)(42,52)(46,56)(47,57)(48,58)(49,59)(53,62)(54,55)(60,64)(61,63);(1,4,7,2)(3,12,15,26)(5,18,22,10)(6,19,8,11)(9,14,17,16)(13,27,29,32)(20,21,25,23)(24,30,31,28)(33,36)(34,40)(35,43)(37,46)(38,47)(39,50)(41,53)(42,54)(44,56)(45,57)(48,61)(49,60)(51,62)(52,55)(58,63)(59,64);(1,5)(2,10)(3,13)(4,18)(6,23)(7,22)(8,21)(9,24)(11,25)(12,27)(14,30)(15,29)(16,28)(17,31)(19,20)(26,32)(33,37,38,49)(34,41,42,48)(35,44,45,59)(36,46,47,60)(39,51,52,58)(40,53,54,61)(43,56,57,64)(50,62,55,63)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157193);