#367607

order	512
nilpotency class	3
isoclinism	isoclinic class of #25140
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#389
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

803 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)(3,7)(4,8)(5,6)(9,10,14,18,15,19,13,27)(11,16,21,31,22,32,28,30)(12,23,26,34,25,38,24,17)(20,35,36,29,37,40,39,33);(2,6)(7,8)(9,11)(10,16)(12,20,25,37)(13,28)(14,21)(15,22)(17,33,34,29)(18,31)(19,32)(23,35,38,40)(24,39,26,36)(27,30);(1,3,5,4)(2,7,6,8)(9,12,13,24,15,25,14,26)(10,17,18,23,19,34,27,38)(11,20,21,36,22,37,28,39)(16,29,30,35,32,33,31,40);(1,4,5,3)(2,7,6,8)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 274803);