#7616409

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, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2399 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1221 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fg,c^2=g,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,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, 7523605);