#7712665

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	#35251	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5698 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)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,26,25,22)(21,28,29,24)(27,30,31,32);(9,11)(10,14)(12,15)(13,16)(17,19)(18,22)(20,26)(21,27)(23,25)(24,30)(28,32)(29,31);(2,6)(3,7)(10,12)(14,15)(18,20)(19,25)(21,29)(30,32);(1,3)(2,4)(5,7)(6,8)(9,12,13,10)(11,15,16,14)(17,20,23,18)(19,26,25,22)(21,28,29,24)(27,32,31,30);(1,4)(2,3)(5,8)(6,7)(9,11)(10,14)(12,15)(13,16)(17,21,23,29)(18,24,20,28)(19,27,25,31)(22,30,26,32);(17,22,23,26)(18,25,20,19)(21,30,29,32)(24,31,28,27)

Polycyclic presentation:

2^9|a^2=h,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, 7619861);