#403128

order	512
nilpotency class	3
isoclinism	isoclinic class of #393906
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1213 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257124, #72257156, #72257160
maximal normal subgroup of	#72257124, #72257156, #72257160

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 310324);