#7312691

order	512
nilpotency class	3
isoclinism	isoclinic class of #7309060
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	#42926	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1531 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=i,c^2=i,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7219887);