#7804784

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

4839 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=gh,d^2=gh,e^2=g,f^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, 7711980);