#408923

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2398
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

815 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 316119);