#6355566

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355552
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	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1565 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)(19,23)(20,30)(21,33)(22,25)(24,31)(26,35)(27,41)(28,37)(29,45)(32,40)(34,39)(36,43)(38,44)(42,48)(46,47);(1,3,6,2,7,9,5,10)(4,11,14,8,12,16,13,15)(17,19)(18,23)(20,27)(21,29)(22,28)(24,36)(25,37)(26,38)(30,41)(31,43)(32,42)(33,45)(34,46)(35,44)(39,47)(40,48);(1,4)(2,8)(3,11)(5,13)(6,14)(7,12)(9,16)(10,15)(17,20,22,32)(18,24,25,39)(19,27,28,42)(21,34,35,31)(23,36,37,47)(26,40,33,30)(29,46,44,43)(38,48,45,41);(4,12)(8,15)(11,16)(13,14)(19,28)(23,37)(27,42)(29,44)(36,47)(38,45)(41,48)(43,46);(1,5,7,6)(2,9,10,3)(4,13,12,14)(8,16,15,11)(18,25)(23,37)(24,39)(26,33)(30,40)(36,47)(38,45)(41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306878, #72306883, #72306925
maximal normal subgroup of	#72306878, #72306883, #72306925

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262762);