#285204

order	512
nilpotency class	3
isoclinism	isoclinic class of #285115
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2424
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3151	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

700 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192400);