#7288294

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234345
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	#42890	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1563 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=fg,d^2=hi,e^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7195490);