#357714

order	512
nilpotency class	3
isoclinism	isoclinic class of #23439
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2626 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(2,4)(5,6)(7,8)(9,10,14,19,15,18,13,28)(11,16,22,33,21,31,29,32)(12,23,26,36,27,17,25,35)(20,30,24,39,37,40,38,34);(1,2,3,4)(9,11)(10,16)(12,24)(13,29)(14,22)(15,21)(17,34)(18,31)(19,33)(20,25)(23,39)(26,37)(27,38)(28,32)(30,35)(36,40);(6,8)(9,12,15,27)(10,17,18,23)(11,20,21,37)(13,26,14,25)(16,30,31,40)(19,36,28,35)(22,38,29,24)(32,39,33,34);(11,21)(16,31)(20,37)(22,29)(24,38)(30,40)(32,33)(34,39)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,g^a=gi,c^b=cg,d^b=di,f^b=fi,g^b=gi,g^c=gi,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264910);