#7699148

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4381 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

107 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7606344);