#346034

order	512
nilpotency class	3
isoclinism	isoclinic class of #345968
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#10238	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

562 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,d^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=def,f^a=fh,g^a=gh,c^b=cg,d^b=de,e^b=eh,f^b=fh,g^b=gi,f^c=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 253230);