#7689749

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	#44980	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4708 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)(10,13)(14,16)(17,18,23,27)(19,28,30,24)(20,25,31,22)(21,26,29,32);(1,3,6,5)(2,4,8,7)(9,10,12,13)(11,14,15,16)(17,19)(18,24)(20,21)(22,26)(23,30)(25,32)(27,28)(29,31);(1,3,6,5)(2,7,8,4)(17,20,23,31)(18,22,27,25)(19,29,30,21)(24,32,28,26);(1,4,6,7)(2,3,8,5)(9,11,12,15)(10,14,13,16)(17,20)(18,25)(19,21)(22,27)(23,31)(24,32)(26,28)(29,30);(1,5,6,3)(2,7,8,4)(17,21)(18,26)(19,20)(22,24)(23,29)(25,28)(27,32)(30,31);(17,22,23,25)(18,20,27,31)(19,26,30,32)(21,28,29,24)

Polycyclic presentation:

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

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596945);