#346539

order	512
nilpotency class	3
isoclinism	isoclinic class of #346518
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#62991	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2220 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=i,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,g^a=gh,e^b=eh,f^b=fi,g^b=gh,e^c=ei,e^d=eh,g^d=gh

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253735);