#7597723

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
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×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1423 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,6)(3,7,4,8)(9,10)(11,15)(12,13)(14,16)(17,18,22,25)(19,26,28,23)(20,21,31,32)(24,27,30,29);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,28)(18,23,25,26)(20,30,31,24)(21,27,32,29);(1,3,5,4)(2,7,6,8)(10,13)(14,15)(17,20,21,25,22,31,32,18)(19,23,29,24,28,26,27,30);(9,11,12,16)(10,14,13,15)(17,19,21,29,22,28,32,27)(18,24,20,26,25,30,31,23);(1,4,5,3)(2,7,6,8)(17,21,22,32)(18,20,25,31)(19,27,28,29)(23,30,26,24)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fhi,d^2=g,e^2=f,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=cgi,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=ei

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, 7504919);