#249796

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	C₂×#41	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

921 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)(4,7)(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)(35,43,45,57)(36,40,47,55)(37,49,50,41)(39,51,54,46)(44,60,61,59)(48,58,52,56)(53,63,64,62);(1,3,7,15)(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)(36,47)(37,50)(40,55)(41,49)(43,57)(44,61)(46,51)(53,64);(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)(33,35)(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);(1,5)(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)(33,36,38,47)(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,c^2=h,d^2=i,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, 156992);