#479951

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

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	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

534 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bg,c^a=ci,d^c=dgi

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72231892, #72321994, #72322004
maximal normal subgroup of	#72231892, #72321994, #72322004

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387147);