#493615

order	512
nilpotency class	4
isoclinism	isoclinic class of #33568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

2347 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)(5,6,11,14,12,17,10,16)(7,18,22,34,23,33,21,35)(8,15,25,31,9,26,24,32)(13,27,28,19,30,36,29,20);(5,7,12,23)(6,13,17,30)(8,20,9,19)(10,22,11,21)(14,28,16,29)(15,33,26,18)(24,27,25,36)(31,35,32,34);(2,4)(6,14)(10,11)(13,28)(15,31)(16,17)(18,34)(21,22)(24,25)(26,32)(27,36)(29,30)(33,35);(5,8,12,9)(6,15,17,26)(7,19,23,20)(10,24,11,25)(13,18,30,33)(14,31,16,32)(21,27,22,36)(28,34,29,35)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^b=di,e^b=ei,f^b=fhi,g^b=ghi,e^c=eh,f^c=fi,g^c=gi,f^e=fi,g^e=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72261075, #72261190, #72261310
maximal normal subgroup of	#72261075, #72261190, #72261310
maximal quotient of	#72261960

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400811);