#162380

order	512
nilpotency class	3
isoclinism	isoclinic class of #162342
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄²×Q₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ 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)(3,10,12,15)(4,13)(5,8)(6,9)(7,11,16,14)(17,18)(19,23)(20,22)(21,24)(25,28)(26,31)(27,29)(30,32)(33,34,36,39,37,40,43,46)(35,38,42,45,41,44,47,48);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(18,22)(20,28)(23,29)(24,25)(27,32)(30,31)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(17,19,21,26)(18,23,24,31)(20,27,28,32)(22,29,25,30)(35,41)(38,44)(42,47)(45,48);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(18,24)(19,26)(22,25)(27,32)

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=gh,e^2=h,f^2=i,b^a=be,d^a=dh,e^a=eg,c^b=ci,e^b=eh,f^b=fgh,d^c=dg

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 69576);