#426020

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
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	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

998 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)(7,9,19,13)(8,21)(11,17,12,27)(14,26)(15,28,20,25)(18,33)(22,35)(23,31)(24,32)(29,30)(34,36);(1,2)(3,4)(5,7,12,15,10,19,11,20)(6,13,27,25,16,9,17,28)(8,18,23,30,24,34,26,35)(14,22,21,33,31,29,32,36);(2,4)(5,8,12,23,10,24,11,26)(6,14,27,21,16,31,17,32)(7,18,15,30,19,34,20,35)(9,22,28,33,13,29,25,36);(5,9)(6,15)(7,17)(8,22)(10,13)(11,25)(12,28)(14,30)(16,20)(18,32)(19,27)(21,34)(23,33)(24,29)(26,36)(31,35)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250313, #72252967, #72256881
maximal normal subgroup of	#72250313, #72252967, #72256881

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333216);