#8096425

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×#546	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5116 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)(11,16)(12,14)(13,15)(17,18,24,28)(19,21,25,20)(22,26,29,31)(23,27,32,30);(1,2)(3,6)(4,7)(5,8)(9,11,13,14)(10,12,15,16)(17,19,24,25)(18,20,28,21)(22,23,29,32)(26,30,31,27);(1,3)(2,5)(4,8)(6,7)(17,20,24,21)(18,25,28,19)(22,30,29,27)(23,26,32,31);(9,11,13,14)(10,12,15,16)(17,21,24,20)(18,19,28,25)(22,27,29,30)(23,31,32,26);(1,3)(2,6)(4,8)(5,7)(9,10)(11,12)(13,15)(14,16)(17,22)(18,26)(19,23)(20,30)(21,27)(24,29)(25,32)(28,31);(1,2)(3,5)(4,7)(6,8)(9,12)(10,14)(11,15)(13,16)(17,23)(18,27)(19,29)(20,26)(21,31)(22,25)(24,32)(28,30)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=g,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

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8003621);