#7941755

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	#46353	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4707 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)(3,6,4,7)(9,10,13,15)(11,12,16,14)(17,18)(19,22)(20,25)(21,24)(23,28)(26,30)(27,32)(29,31);(1,2)(3,6)(4,7)(5,8)(17,18,22,19)(20,23,28,25)(21,24,30,26)(27,31,29,32);(1,3,5,4)(2,6,8,7)(9,11,13,16)(10,14,15,12)(18,19)(23,25)(24,26)(31,32);(1,3,5,4)(2,6,8,7)(9,11)(10,12)(13,16)(14,15)(17,19,22,18)(20,25,28,23)(21,26,30,24)(27,32,29,31);(9,12)(10,11)(13,14)(15,16)(17,20)(18,23)(19,25)(21,29)(22,28)(24,32)(26,31)(27,30);(1,4,5,3)(2,7,8,6)(17,21,22,30)(18,24,19,26)(20,27,28,29)(23,31,25,32)

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7848951);