#7699089

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	#38382	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4583 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606285);