#7763434

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	#37536	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5064 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,c^2=g,d^2=hi,e^2=hi,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,d^c=dh,f^c=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7670630);