#7696397

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	#45355	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5598 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)(3,4)(5,7)(6,8)(10,14)(15,16)(17,18,23,26)(19,20,29,24)(21,22,25,31)(27,30,32,28);(9,10,13,14)(11,15,12,16)(17,19,23,29)(18,24,26,20)(21,27,25,32)(22,28,31,30);(1,3,6,5)(2,7,8,4)(17,20,23,24)(18,19,26,29)(21,30,25,28)(22,27,31,32);(1,4,6,7)(2,3,8,5)(9,11,13,12)(10,15,14,16)(17,20)(18,19)(21,30)(22,27)(23,24)(25,28)(26,29)(31,32);(1,3,6,5)(2,4,8,7)(9,11,13,12)(10,15,14,16)(17,21)(18,22)(19,27)(20,30)(23,25)(24,28)(26,31)(29,32);(1,5,6,3)(2,7,8,4)(9,12,13,11)(10,16,14,15)(17,22)(18,25)(19,28)(20,27)(21,26)(23,31)(24,32)(29,30)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=gh,e^2=gh,f^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, 7603593);