#400116

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	#2598
prime radical	C₂×#2017	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1424 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:

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

Polycyclic presentation:

2^9|a^2=e,d^2=ef,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,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	#72250134, #72257140, #72257218
maximal normal subgroup of	#72250134, #72257140, #72257218

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 307312);