#7678331

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	#44925	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4671 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585527);