#430439

order	512
nilpotency class	3
isoclinism	isoclinic class of #417802
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1208 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:

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

Polycyclic presentation:

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

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, 337635);