#7653425

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	#36434	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4720 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=gh,d^2=g,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dh,c^b=ci,f^e=fhi

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, 7560621);