#8096145

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹⁰×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#2679	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5775 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)(3,7)(4,5)(6,8)(9,10)(11,13)(12,15)(14,16)(18,22)(23,29)(24,30)(31,32);(2,5)(6,7)(9,11,12,16)(10,13,15,14)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30);(1,2)(3,6)(4,5)(7,8)(9,11)(10,14)(12,16)(13,15);(1,3)(2,6)(4,8)(5,7)(9,10)(11,14)(12,15)(13,16)(17,18)(19,23)(20,24)(21,22)(25,31)(26,29)(27,32)(28,30);(17,19,21,26)(18,23,22,29)(20,27,28,25)(24,32,30,31);(17,20,21,28)(18,24,22,30)(19,25,26,27)(23,31,29,32)

Polycyclic presentation:

2^9|b^2=h,e^2=ghi,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,c^b=ci,d^b=dh,f^e=fghi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72211977
maximal normal subgroup of	#72211977
maximal quotient of	#72211974, #72267029, #72267031, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003341);