#7572094

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514020
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1497 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)(9,10,12,14)(11,15,16,13)(17,18)(19,27)(20,32)(21,25)(22,26)(23,30)(24,34)(28,37)(29,38)(31,42)(33,39)(35,45)(36,41)(40,46)(43,48)(44,47);(2,6)(7,8)(9,11,12,16)(10,13,14,15)(17,19,21,30)(18,23,25,27)(20,33,34,28)(22,29,36,31)(24,39,32,37)(26,42,41,38)(35,43,46,44)(40,48,45,47);(1,3,5,4)(2,7,6,8)(10,14)(11,16)(17,20)(18,24)(19,28)(21,34)(22,35)(23,37)(25,32)(26,40)(27,39)(29,44)(30,33)(31,43)(36,46)(38,48)(41,45)(42,47);(1,4,5,3)(2,8,6,7)(10,14)(13,15)(18,25)(19,29)(20,34)(22,36)(23,38)(27,42)(28,43)(30,31)(33,44)(37,47)(39,48)(40,45);(2,6)(7,8)(10,14)(13,15)(18,25)(20,34)(23,27)(26,41)(28,33)(35,46)(38,42)(43,44)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7479290);