#7712367

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	#39725	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4750 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:

(1,2)(3,7)(4,5)(6,8)(10,14)(15,16)(18,24)(22,32)(25,28)(26,30);(9,10)(11,15)(12,16)(13,14)(17,18,23,24)(19,28,20,25)(21,26,31,30)(22,27,32,29);(2,5)(6,7)(9,11,13,12)(10,15,14,16)(17,19,23,20)(18,25,24,28)(21,27,31,29)(22,30,32,26);(1,2)(3,6)(4,5)(7,8)(9,11,13,12)(10,15,14,16)(17,20,23,19)(18,25,24,28)(21,27,31,29)(22,26,32,30);(1,3,4,8)(2,6,5,7)(9,12,13,11)(10,16,14,15)(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|b^2=i,c^2=g,d^2=g,e^2=gh,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, 7619563);