#7918403

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

4979 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)(3,7)(4,5)(6,8)(10,13)(14,16)(17,18,22,25)(19,23,27,21)(20,28,30,24)(26,29,32,31);(1,3)(2,6)(4,7)(5,8)(9,10)(11,14)(12,13)(15,16);(1,2,5,4)(3,7,8,6)(17,19,22,27)(18,21,25,23)(20,26,30,32)(24,29,28,31);(1,3)(2,6)(4,7)(5,8)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,29)(30,32);(1,4,5,2)(3,7,8,6)(17,20,22,30)(18,24,25,28)(19,26,27,32)(21,29,23,31);(1,3)(2,7)(4,6)(5,8)(9,11,12,15)(10,14,13,16)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,28,32)

Polycyclic presentation:

2^9|a^2=hi,c^2=h,e^2=h,f^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7825599);