#7545588

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514000
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7452784);