#421486

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
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

870 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 328682);