#10444946

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

5208 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)(17,18,23,20)(19,24,21,26)(22,25,30,28)(27,31,29,32);(1,3,7,6)(2,4,8,5)(9,10,12,14)(11,13,16,15)(17,19)(18,24)(20,26)(21,23)(22,27)(25,31)(28,32)(29,30);(1,4,7,5)(2,6,8,3)(9,10)(11,13)(12,14)(15,16)(17,18,23,20)(19,26,21,24)(22,25,30,28)(27,32,29,31);(1,4,7,5)(2,3,8,6)(9,11,12,16)(10,13,14,15)(17,20,23,18)(19,26,21,24)(22,25,30,28)(27,31,29,32);(1,5,7,4)(2,3,8,6)(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32);(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=i,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	#72311608
maximal normal subgroup of	#72311608

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10352142);