#403003

order	512
nilpotency class	3
isoclinism	isoclinic class of #400057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1230 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)(5,6,10,15,11,17,9,16)(7,18,21,13,20,12,22,28)(8,23,26,30,24,31,25,14)(19,29,33,32,35,36,34,27);(5,7,10,21,11,20,9,22)(6,12,16,13,17,18,15,28)(8,19,25,34,24,35,26,33)(14,27,23,29,30,32,31,36);(5,7)(6,13)(8,19)(9,22)(10,21)(11,20)(12,15)(14,29)(16,18)(17,28)(23,32)(24,35)(25,34)(26,33)(27,31)(30,36);(2,4)(5,8,9,26,11,24,10,25)(6,14,15,31,17,30,16,23)(7,19,22,33,20,35,21,34)(12,27,28,36,18,32,13,29)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224753, #72224756, #72224758
maximal normal subgroup of	#72224753, #72224756, #72224758

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310199);