#7671633

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4684 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=gi,f^2=g,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, 7578829);