#450592

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	#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

841 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)(5,6,12,28,11,17,10,16)(7,18,27,13,22,34,21,31)(8,14,26,15,23,32,9,24)(19,33,25,30,20,35,36,29);(5,7,10,27,11,22,12,21)(6,13,16,34,17,31,28,18)(8,19,9,25,23,20,26,36)(14,30,24,35,32,29,15,33);(5,8,10,9,11,23,12,26)(6,14,16,24,17,32,28,15)(7,19,27,25,22,20,21,36)(13,29,34,33,31,30,18,35);(2,4)(5,9,10,23,11,26,12,8)(6,15,16,14,17,24,28,32)(7,20,21,25,22,19,27,36)(13,30,18,33,31,29,34,35)

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,c^2=ef,d^2=e,e^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,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250604, #72252977, #72256869
maximal normal subgroup of	#72250604, #72252977, #72256869

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357788);