#432834

order	512
nilpotency class	3
isoclinism	isoclinic class of #418769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,𝔽₄)
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2684
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1550 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:

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

Polycyclic presentation:

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

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, 340030);