#412825

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

982 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250262, #72250279, #72250313, …
maximal normal subgroup of	#72250262, #72250279, #72250313, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320021);