#492254

order	512
nilpotency class	4
isoclinism	isoclinic class of #492252
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×D₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

486 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:

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

Polycyclic presentation:

2^9|b^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,h^a=hi,c^b=cgi,d^b=de,e^b=eg,f^c=fh,h^c=hi,e^d=eg

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72342468, #72342596, #72342602
maximal normal subgroup of	#72342468, #72342596, #72342602

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399450);