#273770

order	512
nilpotency class	3
isoclinism	isoclinic class of #273755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3515	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

737 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)(3,12,15,25)(4,8)(5,10)(6,11)(7,14,9,18)(13,29,31,28)(16,23)(17,24)(19,26)(20,27)(21,30,22,32)(33,34,37,43,39,44,38,42)(35,45,48,56,49,60,47,40)(36,50,53,59,54,58,52,41)(46,55,51,57,62,64,63,61);(1,3)(2,7)(4,13)(5,18)(6,15)(8,21)(9,11)(10,25)(12,27)(14,20)(16,32)(17,31)(19,30)(22,24)(23,28)(26,29)(33,35,39,49)(34,40,44,56)(36,51,54,63)(37,48,38,47)(41,57,59,61)(42,60,43,45)(46,53,62,52)(50,64,58,55);(1,4,6,17)(2,8,11,24)(3,13,15,31)(5,19,20,16)(7,21,9,22)(10,26,27,23)(12,28,25,29)(14,32,18,30)(33,36)(34,41)(35,46)(37,52)(38,53)(39,54)(40,55)(42,50)(43,58)(44,59)(45,61)(47,51)(48,63)(49,62)(56,64)(57,60);(1,3)(2,9)(4,13)(5,18)(6,15)(7,11)(8,22)(10,12)(14,20)(16,32)(17,31)(19,30)(21,24)(23,29)(25,27)(26,28)(33,34,39,44)(35,40,49,56)(36,41,54,59)(37,43,38,42)(45,47,60,48)(46,55,62,64)(50,52,58,53)(51,57,63,61)

Polycyclic presentation:

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

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, 180966);