#7147274

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	#45422	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1835 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=fi,e^2=hi,g^2=i,b^a=bf,c^a=cg,e^a=ehi,g^a=gi,c^b=ch,d^b=df,e^b=eh,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, 7054470);