#8096399

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	C₂×D₄²	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5219 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)(11,12)(13,16)(14,15)(17,18,23,26)(19,27,29,20)(21,24,30,28)(22,25,31,32);(1,2)(3,6)(4,7)(5,8)(9,11,13,15)(10,14,16,12)(17,19,23,29)(18,20,26,27)(21,25,30,32)(22,28,31,24);(1,3)(2,5)(4,8)(6,7)(9,12)(10,15)(11,16)(13,14)(17,20,23,27)(18,19,26,29)(21,22,30,31)(24,25,28,32);(9,12)(10,15)(11,16)(13,14)(17,18)(19,20)(21,24)(22,25)(23,26)(27,29)(28,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,12)(10,11)(13,14)(15,16)(17,21)(18,24)(19,25)(20,22)(23,30)(26,28)(27,31)(29,32);(1,2)(3,5)(4,7)(6,8)(9,10)(11,14)(12,15)(13,16)(17,22)(18,25)(19,28)(20,30)(21,27)(23,31)(24,29)(26,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,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	#72303643
maximal normal subgroup of	#72303643
maximal quotient of	#72303685, #72303692

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003595);