#7631833

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5302 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=h,d^2=g,e^2=g,f^2=h,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539029);