#8088303

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
prime radical	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4944 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=h,d^2=hi,b^a=bg,c^a=ch,e^a=ei,c^b=ci,d^b=di,e^c=ei,f^c=fh,e^d=ehi,f^d=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, 7995499);