#10444954

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

5220 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)(9,10)(11,15)(12,14)(13,16)(17,18,23,26)(19,24,20,21)(22,25,30,32)(27,31,28,29);(1,3,7,6)(2,4,8,5)(9,10,12,14)(11,13,16,15)(17,19,23,20)(18,21,26,24)(22,27,30,28)(25,29,32,31);(1,4,7,5)(2,6,8,3)(9,10)(11,13)(12,14)(15,16)(17,19,23,20)(18,24,26,21)(22,27,30,28)(25,31,32,29);(1,4,7,5)(2,3,8,6)(9,11)(10,13)(12,16)(14,15)(17,20,23,19)(18,24,26,21)(22,27,30,28)(25,29,32,31);(1,5,7,4)(2,3,8,6)(17,21,23,24)(18,20,26,19)(22,29,30,31)(25,28,32,27);(1,6,7,3)(2,4,8,5)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72318542
maximal normal subgroup of	#72318542

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10352150);