#7608092

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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	#28921	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2401 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1371 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fi,c^2=ghi,d^2=gi,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,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7515288);