#7631822

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	#35703	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5174 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^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, 7539018);