#7689150

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	#59818	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5141 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=hi,e^2=ghi,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, 7596346);