#483989

order	512
nilpotency class	4
isoclinism	isoclinic class of #483987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#2218	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

411 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:

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cgh,d^a=dhi,e^a=eg,g^a=gi,d^b=dh,e^b=egi,f^b=fi,g^b=gi,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286070, #72286095, #72286247, …
maximal normal subgroup of	#72286070, #72286095, #72286247, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391185);