#245811

order	512
nilpotency class	3
isoclinism	isoclinic class of #245807
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#61587	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1275 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257603, #72257639, #72257669, …
maximal normal subgroup of	#72257603, #72257639, #72257669, …
maximal quotient of	#72257922

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153007);