#7864184

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
prime radical	#35393	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4853 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=hi,d^2=gh,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7771380);