#481432

order	512
nilpotency class	4
isoclinism	isoclinic class of #481415
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#61427	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₈×#41 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

722 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=g,f^2=h,g^2=i,b^a=be,c^a=ci,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286541, #72286644, #72286648, …
maximal normal subgroup of	#72286541, #72286644, #72286648, …
maximal quotient of	#72343298, #72343323, #72343431, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388628);