#7672619

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	#37839	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4965 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)(3,7)(4,8)(5,6)(10,13)(14,16)(21,29)(24,28)(26,32)(30,31);(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,23,27,25)(21,24,29,28)(26,30,32,31);(2,6)(3,8)(9,11,12,15)(10,14,13,16)(18,20)(23,25)(24,28)(30,31);(1,3,5,8)(2,7,6,4)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(1,4,5,7)(2,8,6,3)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7579815);