#356244

order	512
nilpotency class	3
isoclinism	isoclinic class of #3290
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2361
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1358 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)(5,6,11,15,12,16,10,9)(7,17,21,13,22,26,20,19)(8,23,24,29,25,30,27,14)(18,28,33,31,35,32,34,36);(5,7,10,21,12,22,11,20)(6,13,9,26,16,19,15,17)(8,18,24,34,25,35,27,33)(14,28,23,36,29,32,30,31);(2,4)(5,8)(6,14)(7,18)(9,23)(10,24)(11,27)(12,25)(13,28)(15,30)(16,29)(17,31)(19,32)(20,33)(21,34)(22,35)(26,36);(5,9,10,16,12,15,11,6)(7,19,20,26,22,13,21,17)(8,23,24,29,25,30,27,14)(18,32,33,36,35,28,34,31)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314209, #72314246, #72314266
maximal normal subgroup of	#72314209, #72314246, #72314266

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 263440);