#499152

order	512
nilpotency class	4
isoclinism	isoclinic class of #499151
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2044
commutator, Frattini, cosocle	C₂×#41	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1077 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=e,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,g^a=ghi,c^b=cg,d^b=di,e^b=ei,f^b=fh,g^c=gi,h^c=hi,g^f=gi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303728, #72303740, #72303743, …
maximal normal subgroup of	#72303728, #72303740, #72303743, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406348);