#7744918

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4918 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:

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

Polycyclic presentation:

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

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, 7652114);