#8899529

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

5918 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)(3,4)(5,6)(7,8)(17,18,23,22)(19,25,20,24)(21,26,30,29)(27,32,28,31);(2,6)(3,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,20)(18,24,22,25)(21,27,30,28)(26,31,29,32);(2,6)(4,8)(10,13)(14,16)(17,19,23,20)(18,25,22,24)(21,27,30,28)(26,32,29,31);(2,6)(3,7)(11,15)(14,16)(17,20,23,19)(18,25,22,24)(21,27,30,28)(26,31,29,32);(1,3,5,7)(2,4,6,8)(9,11,12,15)(10,14,13,16)(17,21,23,30)(18,26,22,29)(19,27,20,28)(24,31,25,32);(1,4)(2,7)(3,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,22,23,18)(19,25,20,24)(21,29,30,26)(27,32,28,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=gi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72249526, #72249536, #72249545, …
maximal normal subgroup of	#72249526, #72249536, #72249545, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8806725);