#8438536

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

5099 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8345732);