#7737812

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	#2433	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4812 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)(3,4)(5,6)(7,8)(9,10,13,12)(11,15,16,14)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(1,3)(2,4)(5,8)(6,7)(9,11,13,16)(10,14,12,15);(2,6)(3,8)(9,12,13,10)(11,15,16,14)(18,19)(23,25)(24,26)(30,31);(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3)(2,7)(4,6)(5,8)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4,5,7)(2,3,6,8)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7645008);