#7685075

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

4904 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=h,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7592271);