#7673003

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

5022 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)(3,7)(4,8)(5,6)(10,14)(15,16)(17,18)(19,24)(20,26)(21,22)(23,25)(27,32)(28,29)(30,31);(9,10,13,14)(11,15,12,16)(17,19)(18,24)(20,23)(21,28)(22,29)(25,26)(27,30)(31,32);(2,6)(3,8)(9,11,13,12)(10,15,14,16)(18,25)(19,20)(21,27)(29,31);(1,3,5,8)(2,7,6,4)(17,20,23,19)(18,26,25,24)(21,30,27,28)(22,31,32,29);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,16,14,15)(17,21,23,27)(18,22,25,32)(19,28,20,30)(24,29,26,31);(17,22)(18,27)(19,29)(20,31)(21,25)(23,32)(24,30)(26,28)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,e^2=gh,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,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, 7580199);