#7548478

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1534 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)(3,8)(4,6)(5,7)(9,10)(11,15)(12,13)(14,16)(17,18,21,22)(19,25,27,23)(20,28,30,24)(26,31,29,32);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,23)(20,29)(21,27)(22,25)(24,32)(26,30)(28,31);(1,4,5,3)(2,6,7,8)(10,13)(14,15)(17,20)(18,24)(19,26)(21,30)(22,28)(23,31)(25,32)(27,29);(9,11,12,16)(10,14,13,15)(18,22)(19,25)(20,30)(23,27)(26,31)(29,32);(2,7)(6,8)(17,18,21,22)(19,23,27,25)(20,28,30,24)(26,32,29,31)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=fi,d^2=gi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei,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, 7455674);