#7631079

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	#2847	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5182 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7538275);