#7605401

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgh,c^2=h,d^2=fg,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7512597);