#7968055

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	#35009	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4743 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=gh,d^2=h,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, 7875251);