#6379212

order	512
nilpotency class	3
isoclinism	isoclinic class of #61106
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3586	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,f^a=fi,d^b=df,e^b=ei,f^b=fi,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6286408);