#7804737

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	#59135	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4758 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:

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

Polycyclic presentation:

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

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, 7711933);