#7610790

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	#2407	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1480 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)(4,6)(5,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,21)(20,22)(23,28)(24,27)(25,26)(29,31)(30,32);(1,3,5,4)(2,6,8,7)(9,11)(10,13)(12,15)(14,16)(17,19)(18,25)(20,29)(21,28)(22,30)(23,26)(24,31)(27,32);(1,3,5,4)(2,7,8,6)(10,14)(11,15)(17,20,23,27)(18,22,28,24)(19,29,26,32)(21,31,25,30);(1,4,5,3)(2,6,8,7)(10,14)(13,16)(17,21,22,29,23,25,24,32)(18,26,20,30,28,19,27,31);(2,8)(6,7)(17,22,23,24)(18,27,28,20)(19,30,26,31)(21,29,25,32)

Polycyclic presentation:

2^9|b^2=fi,c^2=f,d^2=fh,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,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, 7517986);