#6379210

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×#3584	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4538 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)(5,6,11,15,12,8,10,23)(7,16,20,13,21,29,19,17)(9,14,26,31,27,22,25,32)(18,33,35,28,36,30,24,34);(5,7,12,21)(6,13,8,17)(9,24,27,35)(10,19,11,20)(14,30,22,33)(15,29,23,16)(18,25,36,26)(28,31,34,32);(2,4)(5,8)(6,12)(7,17)(9,22)(10,23)(11,15)(13,21)(14,27)(16,19)(18,34)(20,29)(24,33)(25,32)(26,31)(28,36)(30,35);(5,9)(6,14)(7,18)(8,22)(10,25)(11,26)(12,27)(13,28)(15,31)(16,33)(17,34)(19,24)(20,35)(21,36)(23,32)(29,30);(5,10,12,11)(6,15,8,23)(7,19,21,20)(9,25,27,26)(13,29,17,16)(14,31,22,32)(18,24,36,35)(28,30,34,33)

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=i,f^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fi,d^b=df,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, 6286406);