#272601

order	512
nilpotency class	3
isoclinism	isoclinic class of #272539
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	#3482	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

763 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)(10,22)(13,25)(15,28)(16,17)(18,24)(21,26)(23,27)(29,30)(31,32)(33,34,40,44,39,46,38,45)(35,43,48,41,49,59,37,55)(36,50,52,42,54,60,53,58)(47,56,51,57,62,64,61,63);(1,3,7,15)(2,9,4,17)(5,21,23,13)(6,16,8,14)(10,26,18,25)(11,28,19,12)(20,32,27,29)(22,30,24,31)(33,35,39,49)(34,41,46,55)(36,47,54,62)(37,40,48,38)(42,56,58,64)(43,44,59,45)(50,63,60,57)(51,53,61,52);(1,4,7,2)(3,12,15,28)(5,18,23,10)(6,19,8,11)(9,14,17,16)(13,29,21,32)(20,22,27,24)(25,31,26,30)(33,36)(34,42)(35,47)(37,51)(38,52)(39,54)(40,53)(41,56)(43,57)(44,50)(45,60)(46,58)(48,61)(49,62)(55,64)(59,63);(1,5)(2,10)(3,13)(4,18)(6,24)(7,23)(8,22)(9,25)(11,27)(12,29)(14,31)(15,21)(16,30)(17,26)(19,20)(28,32)(33,37,39,48)(34,43,46,59)(35,38,49,40)(36,51,54,61)(41,45,55,44)(42,57,58,63)(47,52,62,53)(50,56,60,64)

Polycyclic presentation:

2^9|a^2=fh,b^2=h,c^2=hi,d^2=i,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh

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, 179797);