#7632190

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	#2617	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4946 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,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, 7539386);