#8005627

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

5467 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,d^2=gh,e^2=i,f^2=gh,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

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, 7912823);