#400052

order	512
nilpotency class	3
isoclinism	isoclinic class of #400051
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1365 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307248);