#7729289

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	#37272	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4664 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=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, 7636485);