#7007300

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	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1502 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fi,c^2=i,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,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, 6914496);