#7609932

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	#45002	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=i,c^2=fi,d^2=fhi,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=ei

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, 7517128);