#300303

order	512
nilpotency class	3
isoclinism	isoclinic class of #300269
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62541	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1022 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:

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

Polycyclic presentation:

2^9|b^2=eh,c^2=f,d^2=ei,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=cg,d^b=dh,g^b=gi,d^c=di,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 207499);