#7683357

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	#59409	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4692 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:

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

Polycyclic presentation:

2^9|b^2=gi,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,f^c=fi,e^d=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, 7590553);