#7969448

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	#37832	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5307 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7876644);