#8058950

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	#45680	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4539 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=g,d^2=g,f^2=g,b^a=bg,c^a=ch,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7966146);