#481320

order	512
nilpotency class	4
isoclinism	isoclinic class of #481319
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	#3061	C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ 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) = 256

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321983, #72321989, #72322067, …
maximal normal subgroup of	#72321983, #72321989, #72322067, …
maximal quotient of	#72343615, #72343631

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388516);