#428605

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

819 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)(7,16,20,27)(8,13)(9,14)(10,25)(11,15)(12,19,28,18)(17,32,34,29)(21,35,33,26)(22,30)(23,31)(24,36);(5,7)(6,12)(8,21)(9,18)(10,19)(11,20)(13,29)(14,27)(15,28)(16,25)(17,24)(22,33)(23,34)(26,31)(30,32)(35,36);(5,8,9,23,11,22,10,24)(6,13,14,31,15,30,25,36)(7,17,19,33,20,34,18,21)(12,26,16,29,28,35,27,32);(2,4)(7,18,20,19)(8,22)(12,27,28,16)(13,30)(17,33,34,21)(23,24)(26,29,35,32)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250313, #72252972, #72256905
maximal normal subgroup of	#72250313, #72252972, #72256905

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335801);