#7713881

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4670 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)(3,7,8,4)(9,10,12,14)(11,15,16,13)(17,18,22,19)(20,25,27,23)(21,26,29,24)(28,30,32,31);(1,3)(2,4)(5,8)(6,7)(9,11,12,16)(10,13,14,15)(17,18)(19,22)(20,23)(21,24)(25,27)(26,29)(28,30)(31,32);(2,6)(4,7)(18,19)(20,27)(24,26)(28,32);(1,4)(2,3)(5,7)(6,8)(17,19,22,18)(20,25,27,23)(21,26,29,24)(28,31,32,30);(1,3)(2,4)(5,8)(6,7)(17,20)(18,23)(19,25)(21,28)(22,27)(24,30)(26,31)(29,32);(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)

Polycyclic presentation:

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

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, 7621077);