#253333

order	512
nilpotency class	3
isoclinism	isoclinic class of #253315
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	C₂×C₄²	C₂⁴
prime radical	#18152	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

926 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)(3,5)(4,7)(6,8)(9,10,12,16)(11,18,21,24)(13,22,14,17)(15,20,23,19);(3,8)(5,6)(9,10)(11,18,21,24)(12,16)(13,22)(14,17)(15,20,23,19);(2,5)(3,8)(6,7)(9,11)(10,15)(12,21)(13,20)(14,19)(16,23)(17,24)(18,22);(9,10)(11,15)(12,16)(13,22)(14,17)(18,20)(19,24)(21,23)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285658
maximal normal subgroup of	#72285658

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160529);