#372779

order	512
nilpotency class	3
isoclinism	isoclinic class of #25446
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1407 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)(7,11,17,25)(8,12,19,27)(9,14)(10,13)(15,18,26,23)(16,30)(20,28,33,34)(21,22)(24,32)(29,31)(35,36);(1,2)(3,4)(5,7)(6,11,13,25)(8,16)(9,18)(10,17)(12,24,27,30)(14,26,21,15)(19,32)(20,31)(22,23)(28,36,34,29)(33,35);(2,4)(5,6)(7,15)(8,12)(9,21)(10,13)(11,23)(14,22)(16,29)(17,26)(18,25)(19,27)(20,34)(24,35)(28,33)(30,31)(32,36);(5,8,9,20,10,19,22,33)(6,12,21,34,13,27,14,28)(7,16,18,31,17,32,23,35)(11,24,15,29,25,30,26,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 279975);