#7600912

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	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

1570 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)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,25)(19,28,27,22)(20,31,26,24)(21,32,29,30);(2,5)(3,8)(9,11)(10,13)(12,16)(14,15)(17,19,22,18,23,27,28,25)(20,21,30,31,26,29,32,24);(1,3,4,8)(2,6,5,7)(10,14)(11,16)(17,20,23,26)(18,24,25,31)(19,29,27,21)(22,30,28,32);(2,5)(6,7)(10,14)(11,16)(18,25)(19,27)(20,26)(30,32);(2,5)(6,7)(10,14)(13,15)(17,21,22,31,23,29,28,24)(18,26,27,32,25,20,19,30)

Polycyclic presentation:

2^9|a^2=i,b^2=fgi,c^2=ghi,e^2=fgi,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, 7508108);