#7592055

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518992
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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1373 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18,22,26)(19,27,30,24)(20,23,25,21)(28,32,31,29);(2,5)(6,8)(9,11)(10,13)(12,15)(14,16)(17,19)(18,24)(20,31)(21,29)(22,30)(23,32)(25,28)(26,27);(1,3,4,7)(2,6,5,8)(17,18,23,20,22,26,21,25)(19,28,29,27,30,31,32,24);(11,15)(13,16)(17,20,23,26,22,25,21,18)(19,27,29,31,30,24,32,28);(2,5)(6,8)(17,21,22,23)(18,25,26,20)(19,29,30,32)(24,28,27,31)

Polycyclic presentation:

2^9|a^2=gh,c^2=fgi,d^2=g,e^2=i,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

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, 7499251);