#7712668

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	#37271	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4876 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=i,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619864);