#7685331

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	#2397	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4984 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)(10,14)(15,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(9,10,13,14)(11,15,12,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(1,3,6,5)(2,7,8,4)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31);(1,4,6,7)(2,3,8,5)(9,11,13,12)(10,15,14,16)(17,18)(19,22)(20,23)(21,24)(25,29)(26,30)(27,31)(28,32);(1,3,6,5)(2,4,8,7)(9,11,13,12)(10,15,14,16)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29);(1,5,6,3)(2,7,8,4)(9,12,13,11)(10,16,14,15)(17,20,21,28)(18,23,24,32)(19,25,26,27)(22,29,30,31)

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=ghi,e^2=gh,f^2=gh,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, 7592527);