#420375

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#3054	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1363 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 327571);