#493302

order	512
nilpotency class	2
isoclinism	isoclinic class of #180869
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

591 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=e,e^2=i,f^2=h,g^2=h,h^2=i,b^a=be,c^a=ci,d^a=deh,g^a=gi,c^b=ceh,f^b=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313264, #72313279, #72313298, …
maximal normal subgroup of	#72313264, #72313279, #72313298, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400498);