#7806638

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	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5190 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713834);