#7839616

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	#44877	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5048 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)(9,10,13,12)(11,15,16,14)(17,18,23,24)(19,26,20,21)(22,25,30,31)(27,32,28,29);(9,11,13,16)(10,14,12,15)(17,19,23,20)(18,21,24,26)(22,28,30,27)(25,32,31,29);(2,3)(6,7)(18,24)(19,20)(22,30)(29,32);(1,3,5,2)(4,7,8,6)(17,20,23,19)(18,21,24,26)(22,28,30,27)(25,29,31,32);(1,4,5,8)(2,6,3,7)(9,12,13,10)(11,15,16,14)(17,21)(18,20)(19,24)(22,29)(23,26)(25,28)(27,31)(30,32);(17,22,23,30)(18,25,24,31)(19,27,20,28)(21,29,26,32)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=h,e^2=gh,f^2=i,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,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, 7746812);