#7631770

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	#36411	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=gh,e^2=h,f^2=g,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, 7538966);