#394356

order	512
nilpotency class	3
isoclinism	isoclinic class of #394353
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3021	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2601 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)(5,6,9,12)(7,13,19,25)(8,20,21,30)(10,16,11,15)(14,24,26,22)(17,27,23,29)(18,32,34,36)(28,31,35,33);(6,12)(7,17,19,23)(13,27,25,29)(14,26)(15,16)(18,31,34,33)(20,30)(28,32,35,36);(2,4)(5,7)(6,13)(8,18)(9,19)(10,23)(11,17)(12,25)(14,28)(15,27)(16,29)(20,32)(21,34)(22,33)(24,31)(26,35)(30,36);(5,8)(6,14)(7,18)(9,21)(10,22)(11,24)(12,26)(13,28)(15,20)(16,30)(17,31)(19,34)(23,33)(25,35)(27,32)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72252147, #72256181, #72256186, …
maximal normal subgroup of	#72252147, #72256181, #72256186, …
maximal quotient of	C₂×#394356

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 301552);