#7012741

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	#2408	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1527 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=fi,d^2=i,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=ehi,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6919937);