#479928

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂²×C₄×C₈
prime radical	C₂×#41	C₄²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

562 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)(17,18)(19,23)(20,25)(21,26)(22,24)(27,34)(28,35)(29,33)(30,38)(31,36)(32,37)(39,45)(40,43)(41,44)(42,46)(47,48);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,19,20,27,21,28,30,39,22,29,31,40,32,41,42,47)(18,23,25,34,26,35,38,45,24,33,36,43,37,44,46,48);(3,9)(6,12)(10,15)(13,16)(17,18)(19,23)(20,25)(21,26)(22,24)(27,34)(28,35)(29,33)(30,38)(31,36)(32,37)(39,45)(40,43)(41,44)(42,46)(47,48);(18,24)(23,33)(25,36)(26,37)(34,43)(35,44)(38,46)(45,48)

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,d^a=di,c^b=cg,d^c=di

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231889, #72231892, #72231895, …
maximal normal subgroup of	#72231889, #72231892, #72231895, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387124);