#7689143

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	#59824	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4619 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=hi,e^2=g,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, 7596339);