#7968074

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	#38367	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4788 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)(9,10,12,14)(11,13,15,16)(18,22)(19,25)(24,28)(27,31);(1,3)(2,4)(5,6)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31);(1,4)(2,6)(3,8)(5,7)(17,18,21,22)(19,26,25,23)(20,24,29,28)(27,32,31,30);(1,3)(2,7)(4,8)(5,6)(17,19,21,25)(18,23,22,26)(20,27,29,31)(24,30,28,32);(1,3)(2,4)(5,6)(7,8)(17,18)(19,23)(20,28)(21,22)(24,29)(25,26)(27,32)(30,31);(1,4)(2,6)(3,8)(5,7)(9,11,12,15)(10,13,14,16)(17,20,21,29)(18,24,22,28)(19,27,25,31)(23,30,26,32)

Polycyclic presentation:

2^9|a^2=gi,c^2=h,d^2=h,f^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, 7875270);