#7191142

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996639
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	#44938	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1524 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7098338);