#8093242

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

4870 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=gi,d^2=gh,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000438);