#7712812

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	C₂×#525	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4856 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)(4,6)(5,8)(9,10,12,14)(11,15,16,13)(21,29)(24,28)(26,32)(30,31);(1,3)(2,5)(4,8)(6,7)(9,11,12,16)(10,13,14,15)(17,18,22,20)(19,23,27,25)(21,24,29,28)(26,30,32,31);(2,6)(5,7)(18,20)(23,25)(24,28)(30,31);(1,2)(3,5)(4,6)(7,8)(17,19,22,27)(18,23,20,25)(21,26,29,32)(24,30,28,31);(1,3)(2,5)(4,8)(6,7)(17,20,22,18)(19,25,27,23)(21,28,29,24)(26,31,32,30);(17,21,22,29)(18,24,20,28)(19,26,27,32)(23,30,25,31)

Polycyclic presentation:

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

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, 7620008);