#7846542

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

5074 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:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=h,e^2=gi,f^2=gi,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, 7753738);