#7633150

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=hi,d^2=h,e^2=h,f^2=hi,b^a=bg,c^a=ch,c^b=ci,f^b=fi,e^d=ei

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7540346);