#8598592

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

5897 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261733
maximal normal subgroup of	#72261733

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8505788);