#375941

order	512
nilpotency class	3
isoclinism	isoclinic class of #23994
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2580
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1136 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:

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

Polycyclic presentation:

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

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, 283137);