#7754840

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×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4920 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)(3,4)(5,6)(7,8)(9,10,12,14)(11,15,16,13)(17,18,22,24)(19,25,26,23)(20,29,31,21)(27,32,30,28);(9,11,12,16)(10,13,14,15)(17,19,22,26)(18,23,24,25)(20,30,31,27)(21,28,29,32);(2,6)(3,7)(18,24)(19,26)(21,29)(27,30);(1,3,5,7)(2,4,6,8)(17,19)(18,25)(20,27)(21,28)(22,26)(23,24)(29,32)(30,31);(1,4)(2,7)(3,6)(5,8)(17,20)(18,21)(19,27)(22,31)(23,32)(24,29)(25,28)(26,30);(17,21)(18,20)(19,28)(22,29)(23,30)(24,31)(25,27)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7662036);