#6357812

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1866 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,15)(14,16)(17,18,23,27)(19,28,30,43)(20,25,34,41)(21,26,22,36)(24,31,38,32)(29,42,44,48)(33,39,35,40)(37,46,47,45);(2,6)(7,14)(8,13)(9,15)(17,19,21,31,23,30,22,32)(18,24,36,43,27,38,26,28)(20,29,35,46,34,44,33,45)(25,37,39,48,41,47,40,42);(1,3,5,11,4,10,12,16)(2,7,8,9,6,14,13,15)(17,20)(18,25)(19,29)(21,35)(22,33)(23,34)(24,37)(26,40)(27,41)(28,42)(30,44)(31,46)(32,45)(36,39)(38,47)(43,48);(3,10)(7,14)(9,15)(11,16)(19,30)(24,38)(28,43)(29,44)(31,32)(37,47)(42,48)(45,46);(17,21,23,22)(18,26,27,36)(19,31,30,32)(20,33,34,35)(24,28,38,43)(25,39,41,40)(29,45,44,46)(37,48,47,42)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306887, #72306928, #72306968
maximal normal subgroup of	#72306887, #72306928, #72306968

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265008);