#7744857

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

5366 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gh,d^2=h,e^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, 7652053);