#288628

order	512
nilpotency class	3
isoclinism	isoclinic class of #288611
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#19230	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

665 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=ef,d^2=ei,e^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gi,d^c=di,f^c=fh,g^c=ghi

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, 195824);