#6357763

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357716
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	#62427	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1800 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264959);