#270779

order	512
nilpotency class	3
isoclinism	isoclinic class of #270747
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61947	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1066 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:

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

Polycyclic presentation:

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

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, 177975);