#428400

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1070 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=f,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,f^b=fi,g^b=gi,e^c=ei,g^c=gi,e^d=ei,g^d=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, 335596);