#9203771

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

5389 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72333952
maximal normal subgroup of	#72333952
maximal quotient of	#72334025

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9110967);