#7678466

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	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5054 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)(3,6)(4,5)(7,8)(10,13)(14,16)(17,18)(19,24)(20,21)(22,26)(23,29)(25,31)(27,28)(30,32);(9,10,12,13)(11,14,15,16)(17,19)(18,23)(20,30)(21,28)(22,29)(24,26)(25,27)(31,32);(2,5)(6,8)(17,19,22,29)(18,23,26,24)(20,30,25,27)(21,28,31,32);(1,2)(3,6)(4,5)(7,8)(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,30)(31,32);(17,20,22,25)(18,21,26,31)(19,27,29,30)(23,32,24,28);(1,3,4,7)(2,6,5,8)(9,11,12,15)(10,14,13,16)(17,21)(18,25)(19,28)(20,26)(22,31)(23,27)(24,30)(29,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,e^b=ei,d^c=dh,e^c=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, 7585662);