#7694908

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

4480 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=g,d^2=g,e^2=gi,f^2=i,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, 7602104);