#246413

order	512
nilpotency class	3
isoclinism	isoclinic class of #246367
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	C₂×#538	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1026 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

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

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, 153609);