#7968066

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	#43247	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4756 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)(9,10,13,16)(11,14,12,15)(17,18)(19,22)(20,23)(21,28)(24,26)(25,29)(27,30)(31,32);(1,3)(2,4)(5,6)(7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,24)(20,22)(21,27)(23,26)(25,30)(28,31)(29,32);(1,4)(2,6)(3,8)(5,7)(9,11,13,12)(10,14,16,15)(18,20)(19,26)(21,29)(30,31);(1,3)(2,7)(4,8)(5,6)(17,20,23,18)(19,22,26,24)(21,28,29,25)(27,31,32,30);(1,3)(2,4)(5,6)(7,8)(9,12,13,11)(10,15,16,14)(17,21,23,29)(18,25,20,28)(19,27,26,32)(22,31,24,30);(1,4)(2,6)(3,8)(5,7)(17,22)(18,19)(20,26)(21,30)(23,24)(25,32)(27,28)(29,31)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=h,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, 7875262);