#7654039

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4932 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)(3,8)(4,6)(5,7)(10,13)(14,16)(17,18,23,27)(19,28,30,24)(20,25,31,21)(22,26,32,29);(1,3,5,4)(2,6,7,8)(9,10,12,13)(11,14,15,16)(17,19,23,30)(18,24,27,28)(20,29,31,26)(21,22,25,32);(1,3,5,4)(2,6,7,8)(17,20)(18,25)(19,26)(21,27)(22,24)(23,31)(28,32)(29,30);(1,3,5,4)(2,6,7,8)(9,11,12,15)(10,14,13,16)(17,21,23,25)(18,20,27,31)(19,22,30,32)(24,29,28,26);(1,4,5,3)(2,6,7,8)(17,20)(18,25)(19,29)(21,27)(22,28)(23,31)(24,32)(26,30);(1,2)(3,6)(4,8)(5,7)(17,22)(18,26)(19,25)(20,24)(21,30)(23,32)(27,29)(28,31)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=h,d^2=g,e^2=h,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, 7561235);