#407855

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1007 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)(7,11,17,25)(8,19,20,12)(9,13)(10,14)(15,22)(16,27)(18,24,32,26)(21,29,33,28)(23,31)(30,35)(34,36);(1,2)(3,4)(5,7,9,18,10,17,22,32)(6,11,15,26,14,25,13,24)(8,16,21,30,20,31,33,36)(12,27,29,34,19,23,28,35);(2,4)(5,8)(6,12)(7,16)(9,21)(10,20)(11,23)(13,28)(14,19)(15,29)(17,31)(18,30)(22,33)(24,34)(25,27)(26,35)(32,36);(6,13,14,15)(11,24,25,26)(12,28,19,29)(23,34,27,35)

Polycyclic presentation:

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

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, 315051);