#7600211

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	#30398	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1481 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)(17,18,22,24)(19,27,28,31)(20,26,21,25)(23,29,32,30);(1,3,4,8)(2,5,6,7)(9,10)(11,13)(12,14)(15,16)(17,19,21,30,22,28,20,29)(18,23,26,27,24,32,25,31);(2,6)(3,8)(9,11,12,15)(10,13,14,16)(18,24)(19,28)(25,26)(29,30);(2,6)(5,7)(10,14)(13,16)(17,20,22,21)(18,25,24,26)(19,29,28,30)(23,31,32,27);(2,6)(5,7)(17,19,20,29,22,28,21,30)(18,23,25,31,24,32,26,27)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=h,d^2=i,e^2=fg,f^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,f^a=fi,c^b=cg,d^b=dh,e^b=ei,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, 7507407);