#7560148

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
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1334 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fi,c^2=fi,d^2=fg,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=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, 7467344);