#400264

order	512
nilpotency class	3
isoclinism	isoclinic class of #394464
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	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 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)(5,6,10,16,12,17,11,27)(7,18,21,29,22,33,23,13)(8,14,9,15,26,31,25,32)(19,34,35,28,20,30,24,36);(5,7)(6,13)(8,24)(9,20)(10,23)(11,21)(12,22)(14,30)(15,28)(16,33)(17,29)(18,27)(19,25)(26,35)(31,34)(32,36);(5,8,10,9,12,26,11,25)(6,14,16,15,17,31,27,32)(7,19,21,35,22,20,23,24)(13,28,18,30,29,36,33,34);(2,4)(5,9,10,26,12,25,11,8)(6,15,16,31,17,32,27,14)(7,20,23,35,22,19,21,24)(13,28,33,34,29,36,18,30)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250427, #72252980, #72256910
maximal normal subgroup of	#72250427, #72252980, #72256910

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307460);