#7839622

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

5382 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7746818);