#7696400

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	#36062	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4894 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)(17,18,23,21)(19,20,27,24)(22,25,30,29)(26,28,32,31);(9,10,13,14)(11,15,12,16)(17,19,23,27)(18,24,21,20)(22,26,30,32)(25,31,29,28);(1,3,6,5)(2,7,8,4)(17,20,23,24)(18,19,21,27)(22,28,30,31)(25,26,29,32);(1,4,6,7)(2,3,8,5)(9,11,13,12)(10,15,14,16)(17,20)(18,19)(21,27)(22,28)(23,24)(25,26)(29,32)(30,31);(1,5,6,3)(2,7,8,4)(9,12,13,11)(10,16,14,15)(17,21,23,18)(19,20,27,24)(22,25,30,29)(26,31,32,28);(17,22,23,30)(18,25,21,29)(19,26,27,32)(20,28,24,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=gh,e^2=ghi,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, 7603596);