#7616714

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518952
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	#2540	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#124 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1346 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)(4,8)(5,6)(11,12)(14,15);(2,6)(7,8)(9,10,13,16)(11,15,12,14)(17,18,22,25)(19,26,27,32)(20,24,21,30)(23,29,31,28);(1,3,5,4)(2,7,6,8)(9,11,13,12)(10,14,16,15)(19,27)(23,31)(26,32)(28,29);(9,12,13,11)(10,15,16,14)(17,19,20,29,22,27,21,28)(18,23,30,32,25,31,24,26);(1,4,5,3)(2,7,6,8)(17,20,22,21)(18,24,25,30)(19,28,27,29)(23,32,31,26)

Polycyclic presentation:

2^9|b^2=gi,c^2=fg,d^2=hi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^d=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, 7523910);