#161694

order	512
nilpotency class	3
isoclinism	isoclinic class of #7694
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂²×C₂²⋊C₄
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁

807 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)(3,10,13,7)(4,14)(5,8)(6,9)(11,15,12,16)(17,18)(19,27,29,32)(20,25)(21,26)(22,24)(23,28,31,30)(33,34,36,39,37,40,43,46)(35,38,42,45,41,44,47,48);(1,3)(2,7)(4,11)(5,12)(6,13)(8,15)(9,10)(14,16)(17,19)(18,23)(20,28)(21,30)(22,29)(24,31)(25,27)(26,32)(33,35)(34,38)(36,42)(37,41)(39,45)(40,44)(43,47)(46,48);(1,4,6,5)(2,8,9,14)(3,11,13,12)(7,15,10,16)(18,24)(23,31)(25,26)(27,32)(35,41)(38,44)(42,47)(45,48);(1,5,6,4)(2,8,9,14)(3,12,13,11)(7,15,10,16)(17,20,22,21)(18,25,24,26)(19,28,29,30)(23,27,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 68890);