#404837

order	512
nilpotency class	3
isoclinism	isoclinic class of #393934
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,d^2=g,b^a=be,c^a=cf,d^a=dh,c^b=cg,h^b=hi,h^c=hi,e^d=ei,f^d=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72262159, #72262165, #72262182
maximal normal subgroup of	#72262159, #72262165, #72262182

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 312033);