#7811989

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	#40584	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5184 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7719185);