#7971684

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	#46280	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4757 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)(3,8)(4,7)(5,6)(9,10,13,16)(11,14,12,15)(17,18,22,26)(19,20,24,23)(21,25,31,32)(27,28,29,30);(1,3)(2,4)(5,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,19,22,24)(18,23,26,20)(21,29,31,27)(25,28,32,30);(2,6)(3,7)(19,24)(20,23)(27,29)(28,30);(1,2,5,6)(3,4,7,8)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,29)(31,32);(1,3)(2,4)(5,7)(6,8)(9,11,13,12)(10,14,16,15)(17,20,22,23)(18,24,26,19)(21,30,31,28)(25,27,32,29);(1,4)(2,7)(3,6)(5,8)(9,12,13,11)(10,15,16,14)(17,21,22,31)(18,25,26,32)(19,27,24,29)(20,28,23,30)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,d^2=h,e^2=gh,f^2=gh,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, 7878880);