#10226011

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5760 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)(3,8)(4,7)(5,6)(17,18,22,23)(19,26,25,21)(20,24,28,32)(27,30,29,31);(1,3,5,7)(2,4,6,8)(9,10,13,14)(11,15,16,12)(18,23)(19,25)(20,28)(30,31);(9,11)(10,12)(13,16)(14,15)(18,23)(19,25)(24,32)(27,29);(2,6)(3,7)(10,14)(12,15)(17,19,22,25)(18,21,23,26)(20,29,28,27)(24,30,32,31);(1,2)(3,4)(5,6)(7,8)(9,12)(10,11)(13,15)(14,16)(17,20)(18,24)(19,27)(21,31)(22,28)(23,32)(25,29)(26,30);(1,4)(2,7)(3,6)(5,8)(9,11)(10,15)(12,14)(13,16)(17,21)(18,25)(19,23)(20,30)(22,26)(24,27)(28,31)(29,32)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,d^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72300624
maximal normal subgroup of	#72300624
maximal quotient of	#72300674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10133207);