#7784270

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	#35259	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

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, 7691466);