#8438932

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

5249 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2)(3,8)(4,6)(5,7)(10,12)(14,16)(17,18,21,23)(19,20,24,22);(3,4)(6,8)(11,15)(14,16)(17,19,21,24)(18,22,23,20);(1,3,5,4)(2,6,7,8)(9,10,13,12)(11,14,15,16)(17,19)(18,20)(21,24)(22,23);(1,4,5,3)(2,6,7,8)(9,11,13,15)(10,14,12,16)(17,20)(18,19)(21,22)(23,24);(9,12,13,10)(11,16,15,14);(1,2)(3,6)(4,8)(5,7)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=h,e^2=ghi,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, 8346128);