#7940919

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	#2388	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4479 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7848115);