#493878

order	512
nilpotency class	4
isoclinism	isoclinic class of #480274
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×H(5,𝔽₂)	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂²×H(5,𝔽₂) ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1831 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)(3,6)(4,5)(7,8)(9,10,15,16,13,19,11,21)(12,18,23,17,14,20,22,24);(2,5)(6,8)(9,11,13,15)(10,16,19,21)(12,23,14,22)(17,18,24,20);(1,3,4,7)(2,6,5,8)(9,11,13,15)(10,17,19,24)(12,22,14,23)(16,18,21,20);(9,12,13,14)(10,18,19,20)(11,22,15,23)(16,17,21,24)

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=e,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,f^a=fh,h^a=hi,d^b=di,f^c=fi,g^c=ghi,g^f=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72232794, #72233002, #72233095, …
maximal normal subgroup of	#72232794, #72233002, #72233095, …
maximal quotient of	#72232783, #72233221, #72260536

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401074);