#249797

order	512
nilpotency class	3
isoclinism	isoclinic class of #249759
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	#3106	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

935 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)(4,18)(5,21)(6,10)(7,11)(8,15)(9,19)(13,17)(14,23)(16,24)(20,25)(22,26)(27,29)(28,31)(30,32)(33,34,38,42)(35,43,45,57)(36,40,47,55)(37,49,50,41)(39,51,54,46)(44,60,61,59)(48,58,52,56)(53,63,64,62);(1,3,7,17)(2,8,11,24)(4,14,20,30)(5,16,6,15)(9,23,26,32)(10,13,21,12)(18,27,25,28)(19,31,22,29)(36,47)(37,50)(40,55)(41,49)(43,57)(44,61)(46,51)(53,64);(1,2)(3,13)(4,9)(5,10)(6,21)(7,11)(8,15)(12,17)(14,28)(16,24)(18,19)(20,26)(22,25)(23,31)(27,30)(29,32)(33,35)(34,39)(36,44)(37,51)(38,45)(40,53)(41,57)(42,54)(43,49)(46,50)(47,61)(48,63)(52,62)(55,64)(56,59)(58,60);(1,4,7,20)(2,9,11,26)(3,14,17,30)(5,22,6,19)(8,23,24,32)(10,25,21,18)(12,27,13,28)(15,31,16,29)(33,36,38,47)(34,40,42,55)(35,44,45,61)(37,52,50,48)(39,53,54,64)(41,58,49,56)(43,59,57,60)(46,63,51,62)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=di,e^b=eh,f^b=fi,g^b=gh,e^c=ei,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, 156993);