#7689727

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	#44743	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4750 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gh,e^2=h,f^2=g,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, 7596923);