#411065

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

821 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=gh,c^2=fg,d^2=f,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,e^d=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 318261);