#7883032

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	#45002	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4755 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)(5,6)(7,8)(10,14)(15,16)(17,18,23,21)(19,27,30,24)(20,25,29,31)(22,26,28,32);(1,3,7,5)(2,4,8,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,24)(20,28)(21,27)(22,29)(23,30)(25,26)(31,32);(1,4,7,6)(2,5,8,3)(17,18,23,21)(19,27,30,24)(20,25,29,31)(22,26,28,32);(1,3,7,5)(2,4,8,6)(9,11,13,12)(10,15,14,16)(17,20,23,29)(18,25,21,31)(19,28,30,22)(24,26,27,32);(1,5,7,3)(2,4,8,6)(9,12,13,11)(10,16,14,15)(17,21,23,18)(19,27,30,24)(20,31,29,25)(22,26,28,32);(1,6,7,4)(2,5,8,3)(17,22,23,28)(18,26,21,32)(19,29,30,20)(24,25,27,31)

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7790228);