#453998

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

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

2234 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:

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

Polycyclic presentation:

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

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, 361194);