#146696

order	512
nilpotency class	4
isoclinism	isoclinic class of #12711
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×#41	C₂³
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#520 ◁ G
LECS	G ▷ C₂²×#41 ▷ C₂² ▷ C₂ ▷ C₁

1094 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(5,6)(7,17)(8,21,9,14)(10,15,25,31)(11,16)(12,18,28,23)(13,22)(19,26,34,36)(20,30)(24,29)(27,35)(32,33);(1,2,3,4)(5,7)(6,12)(8,18)(9,23)(10,19)(11,20)(13,26)(14,30)(15,27)(16,28)(17,21)(22,33)(24,35)(25,34)(29,36)(31,32);(5,8,11,9)(6,13,16,29)(7,18,20,23)(10,22,25,24)(12,26,28,36)(14,15,21,31)(17,32,30,27)(19,33,34,35)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,h^a=hi,d^b=di,f^b=fhi,f^c=fh,f^d=fi,f^e=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53892);