#242673

order	512
nilpotency class	3
isoclinism	isoclinic class of #10603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#396
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#396
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

683 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=i,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,g^a=gh,c^b=cg,g^b=gh,e^c=eh,f^c=fi

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, 149869);