#7590508

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518920
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

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

1413 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=f,c^2=gi,e^2=fi,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,d^c=di

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, 7497704);