#7806698

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,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, 7713894);