#433014

order	512
nilpotency class	3
isoclinism	isoclinic class of #425091
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

927 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340210);