#492125

order	512
nilpotency class	4
isoclinism	isoclinic class of #491728
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
prime radical	C₂×D₁₆	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

400 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(2,6)(4,12)(7,15)(8,13)(9,16)(10,14)(17,18,23,28)(19,24,32,40)(20,26,22,36)(21,25,35,42)(27,33,41,34)(29,38,31,46)(30,37,45,48)(39,43,47,44);(1,2,5,8)(3,9,11,15)(4,13,12,6)(7,14,16,10)(17,19,23,32)(18,24,28,40)(20,29,22,31)(21,30,35,45)(25,37,42,48)(26,38,36,46)(27,39,41,47)(33,43,34,44);(18,25)(20,22)(21,34)(24,37)(26,27)(28,42)(29,31)(30,44)(33,35)(36,41)(38,39)(40,48)(43,45)(46,47);(1,3,5,11)(2,7,8,16)(4,14,12,10)(6,15,13,9)(17,20,23,22)(18,26,28,36)(19,29,32,31)(21,33,35,34)(24,38,40,46)(25,41,42,27)(30,43,45,44)(37,47,48,39)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287679
maximal normal subgroup of	#72287679

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 399321);