#7699062

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	#43907	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5091 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)(4,8)(5,6)(10,14)(15,16)(17,18)(19,27)(20,25)(21,26)(22,24)(23,28)(29,31)(30,32);(9,10,13,14)(11,15,12,16)(17,19)(18,23)(20,29)(21,25)(22,28)(24,27)(26,30)(31,32);(2,6)(3,8)(9,11,13,12)(10,15,14,16)(18,24)(19,28)(25,32)(26,29);(1,3,5,8)(2,7,6,4)(9,11,13,12)(10,15,14,16)(17,19)(18,23)(20,29)(21,25)(22,28)(24,27)(26,30)(31,32);(1,4,5,7)(2,8,6,3)(9,12,13,11)(10,16,14,15)(17,20)(18,25)(19,29)(21,23)(22,30)(24,32)(26,28)(27,31);(17,21)(18,26)(19,25)(20,27)(22,31)(23,30)(24,29)(28,32)

Polycyclic presentation:

2^9|b^2=gi,c^2=gi,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, 7606258);