#450932

order	512
nilpotency class	3
isoclinism	isoclinic class of #417542
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁵ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1417 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358128);