#7698916

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×#553	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4560 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606112);