#7633310

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	#35078	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5376 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)(11,16,15,12)(17,18,22,26)(19,20,24,23)(21,25,30,29)(27,28,32,31);(9,11)(10,12)(13,16)(14,15)(17,18,22,26)(19,20,24,23)(21,29,30,25)(27,31,32,28);(1,2,4,5)(3,6,7,8)(9,12,14,16)(10,15,13,11)(17,19)(18,23)(20,26)(21,27)(22,24)(25,31)(28,29)(30,32);(9,10,14,13)(11,12,15,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,31)(30,32);(17,20)(18,24)(19,26)(21,28)(22,23)(25,32)(27,29)(30,31);(1,3,4,7)(2,6,5,8)(9,13,14,10)(11,16,15,12)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,32)(26,29)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,c^2=hi,d^2=g,f^2=hi,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, 7540506);