#7631127

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

5168 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=gh,d^2=h,e^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, 7538323);