#7697244

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	#39416	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4814 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=gh,e^2=hi,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, 7604440);