#7734330

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	#37673	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5077 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=g,e^2=g,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, 7641526);