#6372018

order	512
nilpotency class	3
isoclinism	isoclinic class of #60457
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂×#3584	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6279214);