#8075453

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	#44161	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4620 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=hi,d^2=h,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=ei,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, 7982649);