#7712749

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

4656 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,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, 7619945);