#6354821

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354774
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60790	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,7,10)(3,5,9,6)(4,8,12,15)(11,13,16,14)(17,18,23,28)(19,29,31,40)(20,34,35,42)(21,27,22,26)(24,33,39,32)(25,37,41,36)(30,44,45,38)(43,47,48,46);(1,3,6,2,7,9,5,10)(4,11,14,8,12,16,13,15)(17,19,21,32,23,31,22,33)(18,24,26,29,28,39,27,40)(20,30,36,46,35,45,37,47)(25,38,34,43,41,44,42,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,12)(9,16)(10,15)(17,20)(18,25)(19,30)(21,36)(22,37)(23,35)(24,38)(26,34)(27,42)(28,41)(29,43)(31,45)(32,46)(33,47)(39,44)(40,48);(4,12)(8,15)(11,16)(13,14)(19,31)(20,35)(24,39)(25,41)(29,40)(32,33)(34,42)(36,37);(1,5,7,6)(2,9,10,3)(4,13,12,14)(8,16,15,11)(19,31)(24,39)(29,40)(30,45)(32,33)(38,44)(43,48)(46,47)

Polycyclic presentation:

2^9|a^2=h,b^2=f,e^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,d^b=di,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306988, #72307011, #72307064
maximal normal subgroup of	#72306988, #72307011, #72307064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262017);