#7732544

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	#35568	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4504 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=g,d^2=g,e^2=i,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, 7639740);