#7603275

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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1598 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)(3,7)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18,22,27)(19,30,32,25)(20,34,33,42)(21,36,38,45)(23,29,24,28)(26,40,44,39)(31,43,47,48)(35,46,37,41);(2,6)(7,8)(10,13)(11,16)(17,19,22,32)(18,25,27,30)(20,23,33,24)(21,37,38,35)(26,43,44,48)(28,42,29,34)(31,39,47,40)(36,46,45,41);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,23,32,22,33,24,19)(18,25,28,34,27,30,29,42)(21,35,39,47,38,37,40,31)(26,41,36,48,44,46,45,43);(1,4,5,3)(2,8,6,7)(10,13)(14,15)(17,21,24,40,22,38,23,39)(18,26,29,45,27,44,28,36)(19,31,33,37,32,47,20,35)(25,41,42,43,30,46,34,48);(2,6)(7,8)(10,13)(14,15)(18,27)(21,38)(25,30)(28,29)(31,47)(34,42)(35,37)(39,40)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=fh,d^2=fg,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, 7510471);