#7905682

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	#37644	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7812878);