#8077323

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

4897 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7984519);