#7741183

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	#35251	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5408 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)(3,7)(4,5)(6,8)(9,10,13,15)(11,16,14,12)(19,25)(23,29)(26,31)(30,32);(9,11)(10,12)(13,14)(15,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(2,5)(6,7)(9,12,13,16)(10,14,15,11)(18,22)(19,25)(24,27)(26,31);(1,2)(3,6)(4,5)(7,8)(9,12)(10,11)(13,16)(14,15)(17,19,21,25)(18,23,22,29)(20,26,28,31)(24,30,27,32);(1,3)(2,6)(4,8)(5,7)(9,11)(10,12)(13,14)(15,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7648379);