#8096309

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

5198 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)(11,12,15,14)(17,18)(19,20)(21,29)(22,26)(23,24)(25,31)(27,32)(28,30);(1,2)(3,6)(4,7)(5,8)(9,11)(10,14)(12,16)(13,15)(17,19)(18,23)(20,26)(21,30)(22,24)(25,28)(27,31)(29,32);(1,3)(2,5)(4,8)(6,7)(17,20,22,23)(18,24,26,19)(21,28,31,32)(25,30,29,27);(9,11)(10,14)(12,16)(13,15)(17,21)(18,25)(19,27)(20,28)(22,31)(23,32)(24,30)(26,29);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,15)(14,16)(17,19)(18,20)(21,27)(22,24)(23,26)(25,28)(29,32)(30,31);(1,2)(3,5)(4,7)(6,8)(9,12)(10,15)(11,16)(13,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,32)(28,30)(29,31)

Polycyclic presentation:

2^9|a^2=gh,c^2=h,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	#72304025
maximal normal subgroup of	#72304025
maximal quotient of	#72304103, #72304137

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003505);