#7685690

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4971 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=h,e^2=ghi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592886);