#7896033

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

4725 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=g,d^2=gi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7803229);