#225501

order	512
nilpotency class	3
isoclinism	isoclinic class of #25440
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	D₄×C₄⋊C₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

610 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=fh,e^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,d^b=dei,e^b=eh,e^d=eh

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, 132697);