#7805059

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

4974 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,d^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,e^b=ei,f^b=fi,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, 7712255);